Basis properties for inverse semigroups

,ounivAL

OF

ALGEBRA

50,

135-152

(1978)

Basis Properties

for

Inverse

P. R. Depurttnevt

of Mathematics,

University

Semigroups*

JONES

of Glasgow,

Glasgow

G12 8QW, Scotland

Conmur~icated by G. B. Preston Received

November

26, 1976

In a previous paper [9], the author showed that a wide class of inverse semigroups (for example free inverse semigroups, semilattices and finite combinatorial inverse semigroups) possess the strong basis property, and thus the basis property, which we now define. If S is an invcrsc semigroup and C < C; ::z S, a CT-basis for I’ is a subset S of V, minimal such that (Z! u .Y) 7: V, and a hasis for V is an O-basis for V, S is said to have the basis property [strong basis property] if for any inverse subsemigroup V of S [and inverse subsemigroup I,’ of I’] any two bases for C’ [U-bases for I’] have the same cardinality. (One notable consequence of the strong basis property for S is that for any j’-class J of S, any two U-bases for V have the same number of elements in J [9, Theorem 3.91). Two questions were put in [9]:

(I) 1Vhat are necessary and sufficient conditions for an inverse to have (a) the strong basis property, or (b) the basis property? (II) Are the basis inverse semigroups ?

property

and the strong

basis property

semigroup

equivalent

for

To answer (I)(a), we show first (in Sect. 2) that an inverse semigroup has the strong basis property if and only if each of its principal factors has, (but this is not true for the basis property, see Sect. 6). The main result then follows by finding those Rrandt semigroups with the strong basis property. ‘L’IIEOREM

(Theorem

4.8).

An inverse

semigroup

has the strong

hasi.s prope’pYtQ’

if rind on& if:

- The author wishes to thank N. K. Dickson and M. J. ‘l‘omkinson for their advice and helpful suggestions on the group theoretic nspects of this work, the former in Section 5 and the latter in Section 4, in particular. The main results above were first presented in an invited lecture at the University of St Andrew, Scotland, on Sovember 20, 1975.

I35 OOZI-8693/78/0501-0135$02.00/O Copyright 0 1978 by Academic Press, Inc. ..\I1 rights of reprodwtion in any form reserved.

48r/jo/r-10

136

I’.

rind

(i)

it is completely

11

each non-isolated

R. joses

semisimple; maximal

subgroup

is a primavy

iY-gJ,oap;

(‘I

(iii)

each isolated

maximal

subgroup

has

the strong

basis propert?*.

(.% maximal subgroup of an inverse semigroup is isolated if it constitutes a whole P-class, and otherwise nonisolated. A group G is an AT-g~oup [IO] if for any subgroups C: and I7 of G, C! maximal in I’ implies 1. normal in I-.) For example a finite inverse semigroup in which each subgroup is primar!. has the strong basis property. The basis property is less amenable to inductkc treatment. IHowever WC can show (‘Theorem 6.1) that a Rrandt semigroup P(G; I, I; -1) has the basis property if and only if G does. ‘This enables us to answv (II) in the negative. In Section 5, finite groups arc considered briefly. \Ye show, for example (Theorem .5.4), that any finite group with the basis property is soluble.

I. INTR~DUCTIOI\‘ In general, we will use the notation and definitions of [9], except that in order to avoid confusion with commutators when considering groups, the notation .\- (rather than [-VI) will b e used to denote the inverse suhsemigroup generated by the set S. For elementary properties of inverse semigroups, the reader is referred to [3] and for groups to [I I]. \Ve retain the convention that the empty set C is an (inverse) semigroup, and thus an inverse subsemigroup of any inverse semigroup, except in the cast will have its usual meaning (that is, of groups, where the term “subgroup” “in~rse subsemigroup containing the identity”). ,Y (that is, L’ is Recall from [9] that if S is an inverse scmigroup and I. an inverse subscmigroup of S), a subset S of S is said to be i .-iuredundant S, a 1 .-basis for I _is a I’-irredundant if .Y$ (1.) .Y’s’ , .I’ t S. Thus if li c I7 I ‘, and a basis for I ’ is an iwedundant subset S of I. such that ,‘C’, S,~ (that is. r;-irredundant, or {e}-irredundant if S is a group) subset S such that _.IJ *. ‘To begin, some of the results of [9] arc summarized. ‘l’he most important of these is the following, which is essentially contained in the proof of Theorem 2.3 of [9]. &xu.T

elements at5 :z:,j-

1 .I. i,et s be an inverse semigroup s, J$, a of S and U 5.: S, \I L-‘, x, ~1: Then ,C; has the strong basis propert?,.

that for ally ,y-related <‘CT, a)> implies a E ’ Zi, S, 01

such

I37

BASIS PROPERTIES FOR ISVERSE SERlIGROllPS

An inverse semigroup S is said to have the generating propevty if an)- two bases for S haye the same cardinality. Thus S has the basis propert!- if and only if each of its inverse subsemigroups has the generating property. From cardinality arguments, any inverse semigroup which cannot be finitely generated has the generating property. So in fact S has the basis property if and only if each of its finitely generated inverse subsemigroups has the generating propert!.. However as there is no result analogous to Result 1.I, the basis propert!- is less amenable than the strong basis property. RESULT 1.2 19, Theorem is ronlpleteh

5.11.

.4n

inverse

[email protected]

with

the

Basis

propp1’tJT

semisimple.

In this paper, then, we shall be considering only completely semisimple inverse semigroups, some of whose properties are rcriewed here for reference. If J is a j-class of the inverse semigroup S, the principal factor Z’F(J) is defined Sd! as the Rees factor semigroup J(a)/I(a), f or some (any) a E J, where J(a) and I(N) J(a)‘J. (Thus if I(a) = 0, D’(J) :-- J). An inverse semigroup .C is conzp/ete/y semisimple if each principal factor is completcl>- simple or completely O-simple, that is [9, Theorem 3.91, a group or a Hrandt semigroup rrspectively. RIX:LT

1.3 [9, Theorem

The ~foliowi7~,y

nre

1.I].

I,et S he a12ineerse sem<~foup.

equivalent:

(i)

S is complete~l~ semisimple;

(ii)

‘9 f-l - ,. (.5)

(iii)

.9n.,s

(IV)

(v)

clb.Vn

~‘OROI.I.XRY (i)

(ii)

1; implies

ah9a9b

Fus ther 9

1;

ah&u

(or

the dual:

ahpib

inlpbes

crM’b),

(N, 0 E S);

implies a.Fb ‘, (a, b E S).

,p in an\1 complete!\! semisimple irie~esse.wm@vup. I .4.

r e E(S),

!f

S is a complete!\.

.s E S ad

es~~~x[se~~~?c]

semisimple together

inverse hnpfj~

senuqr-oup, es

.\:\~.Y//.Yimplies .~~\~sX”s; q’.v ‘P’s implies .\:\‘.I’ ‘Xs.~

.x [se

then: ~= x] :

I, (,v, -1’6 S).

Part (i) of this corollary will bc used continually in the following ways. If .\ /tez’, where .s, u, z’ E S and e E E(S), and if x9 21, then zce,fu, so ur II, that is .\ W. (\I-e say the idempotent factor e is suppress4 from the expression for s.) .\lso, if .x uav, where a E S and U, z E S, and if .v,p a, then II ‘.\.e. ’ (u -‘u) u(m-*) a. In viclv of Result I .I and the comments after it, the existence of bales (or

138

P. K. JOluES

U-bases) is somewhat irrelevant to our discussion. To say an inverse semigroup has the generating property, for instance, is not to imply that it necessarily has a basis. In [9] it was shown, however, that many infinitely generated inverse semigroups (in particular all the infinitely generated inverse subsemigroups of a free inverse semigroup) do indeed have a basis.

We begin the proof of our main result by show-ing that an inverse semigroup has the strong basis property if and only if each principal factor has. (This immediately yields answers to Questions (I) and (II) for com~nutatk inverse semigroups, for instance.) W-e also show that the strong basis property is preserved by idempotent-separating homomorphisms of inverse semigroups. Some definitions and notation are needed. Let S be an inverse semigroup with E B(S) and b(S) its poset of (y-classes (where Jc .i Ji, itf SuS C SbS, N, h E S). If n E 5’ and tT .-C S, define I-,, [Ue n J,, I e E E 17 Jc,)‘,. Clear11

Pmof. Suppose0 ZLjUZC~ ) where zcr , w.,- E ‘C!, .S .l and u E C’. ‘I’hen \ 0 w,lLx,(u~ ‘a) zLj(u(w2a- lazcyl)) wpu -%2 U~U’W~ (by C‘orollary I .4(i)), where u’ u(w,a *a&) E U, , since w,a -%zw$ E E n JO . Thus a E
brisis

Proof. Suppose S has the strong basis property. Let I be an ideal of .V and 1. an inverse subsemigroup of S/I, and suppose S and A4 are 1,.-irredundant ( I,‘, .-1 but S ; .-I ‘. \I’e must obtain subsets of s/r such that ,‘V, .Y’ 3 contradiction. If S contains I (the zero of S/Z), then either I & .1 (whence I E I r, -4 ‘I’, =?:.I‘,, with .%I c I,‘-,S:I ) a contradiction) or 1~ A and ‘I-, s’,I .-I\\1 ~. -1 similar result is obtained if I t -4. \Vithont loss of generality, then, we may assutne that 1 4 a\- U .-I. Denote by q the natural homomorphism (injectire on S\,I) of S upon $!Z. Then S’ I-7 ’ ; -\;I-l and z4’ ~~.4~ 1 are subsets of S,J, with ~X’ ! .:- A’ ‘. Put I *’ (by the injectivity then i I-‘, &Y’, ,I.‘, :I ) and -Y’, .-I’ are I”-irredundant of 1 on S’,I). Rut this contradicts the strong basis proper% for S. Hence S’I has the strong basis property.

BASIS

[I 13~ putting 1. property is preserved.

PROPERTIES

throughout, m

FOR ISVERSE

SEMIGROUPS

the above argument

139

also shows the basis

(By arguments similar to those above and Lemma 3.7 of [9], it can be shown that if S/I has the strong basis property for every ideal 1 of S, then 5’ has the strong basis property, but this is not needed in the sequel.) Now since every principal factor of a semigroup S is a Rees factor semigroup of an ideal of S, it is clear from the abore lemma that if an inverse setnigroup has the strong basis property, then so does cl-cry principal factor. 2.3. Let S be an iwuerse semigroup. Then S has the strong basis if and only if each principal factor has.

THEOREM

pyopwty

Proof Only sufficiency remains to be proved. Suppose each principal factor of S has the strong basis property. By Result 1.2, each factor is completely semisimple and thus so is S. By Result I. 1, to show S has the strong basis property we need only consider /-related elements s, y, a of S, and 0’ < S such that (U,x,y) =- , and show that either a E (11, N) or UE(U,a. But y E (U, a>, so y E (U u E(S), rj. Also n E (U, X, y); therefore either Q E or every expression for a as a product in (U, ~,y) involves ( I/‘, x), a E (U,y) .r and y. Then substitute for y, and suppress idempotent factors (using Corollary 1.4(i)), giving a E (U, ~‘1, contradicting our assumption. m The analogous result for the basis property is false, as Example 6.3 will show, although by Proposition 2.2, necessity remains true. A completely semisimple inverse semigroup S has at most one group principal factor, that corresponding to the kernel of S, if one exists. The remainder are Brandt semigroups, of which those with just one nonzero idempotent arc groups with zero (and clearly have the [strong] basis property iff the same is true of the group itself). Those Brandt semigroups which are not groups with zero will be considered in the next section. Some conclusions can be immediately drawn from Theorem 2.3, however. COROLLARY 2.4. A semilattice of groups has the strong basis property only if each gvoup has.

if and

Kow it is well known that any commutative inverse semigroup is a semilattice of Abelian groups. It will be shown in Section 4 that for an Abelian group A,

the basis propel-t!- and the strong basis propert!’ are each equivalent to ! being primq (that is. a p-group for some prime p). ‘I’hus questions (I) and (II) are answered in this case hy the following.

(i)

,Y Iius the basis property’;

(ii)

fr’erj’ (ir2nsimnl) subgroup of S is prirnnr-11;

(iii)

,Y ho5 the strop

basis property,.

I:urthcr discussion of the 1x4s propertics for groups is contained in Sections 4 and 5. Of particular use in the special case of a group homomorphism is the following result. THEOREM 2.6. The strorg separating komomovphisms.

basis proper?\!

is preserzqed under idempotent-

Proof. Let S and 7’ be inverse semigroups with 4: S -+ 1’ a surjectivc, Note first that if sPt[sSt] in T, and idempotent-separating homomorphism. Therefore s2’t implies s,Yt, $14 ~- $1 tf$ f, (5 , t, E S) then slYt,[s,BV,]. e,d) and f’ also. Suppose e, f E E( I’) and (e,f) E P n . Pick x1 E ~5, ~1,y1 E&I and a, E a~$ I; thus x1 . ~1~, and a, are in the same 2’ (~ Q)-class of S. Let K (s E S 1sd, E E(T):, the kernel of 4. Now x1$ x E 1I’, a?, so x1+ = a$ for some 20E TV+‘, al>>. Therefore (xlwml)$ =~ (ZW-‘)$, that is s~F:’--l E K. Further x1 (x17r1)zc (since (W1w)+ =: (.+&), so ml E :; CT,(I~\,, ,/ V4-l u K>. Similarly 3~~E (L,‘, a,) and a, E (C;, .q , yl?,. Thereforc where C ‘{CT, a,> in S, and so we may assume, without loss of generality, : L, x1 ) j’l that a, E (U, x1:‘. 1Ve now show that this implies n E ‘I’, s\ or a E {I’, J“,, from which the theorem follows, by Result 1.1. Now ~2,E (U, .x1>, implies a E . But a E ( V, s, J?>, so tither a E ( V, xi, doE :I-, JV‘, or every expression for (2 in (,‘C’, s, JC> involves both x and y. In this last case, however, by substituting for )‘, and suppressing idempotent factors, we have a E ‘V, 1 \ .A+),a contradiction. COROLLARY 2.7. homomorphic image.

If

a group has the strong basis property,

so does ez~er~;~

BASIS PROPERTIES FOR INVERSE W~IIGROCPS

141

That this result is not true in general for inverse semigroups is clear, since free inverse semigroups have the strong basis property [9, Corollary 2.51, but not all inverse semigroups do, by Result 1.2.

3. RRANOT SEMIGR~UPS From Theorem 2.3 and the remarks following it, we need novv to find those Brandt semigroups with the strong basis property, in fact only those which are not groups with zero (that is, those which have at least two nonzero idempotents). \\‘e prove the following theorem. THEOREM 3.1. Let S be a Brandt semigroup which is not a group with zero, and let G be a maximal (nonzero) subgroup. Then S has the strop basis propert ;f and only if G is an n-group with the strong basis propertJq.

Here an Av-gvoup ([lo, p. lo]) is a group such that for any subgroups Land I ., 1. a masimal subgroup of V implies U is normal in L7. (This propert! was introduced by Baer [I], who called it (G).) The following lemma gives LIS an alternative characterization of iy groups. (If L ::i G and
3.2. .-I group G satisfies iv if and on& ij it satisfies

(.I:

ijL

z< G, g E G and g E ;~L, Lg, , then g EL.

Proof If G is an ~%rgroup, suppose I, < G, g E G, ,y E ‘::I,, 1,” and g 6 I,. By Zorn’s lemma, there exists a subgroup ,I1 of (L,g, , maximal subject to containing L but not g. Then M is maximal in , then L .;: .Y and g E AT, so N _ ,‘L,g, . So M 4 (‘L.g;. Then, however, %qE ‘L, I,!‘, I ~iz2,&IQ) == AI, a contradiction. Therefore G satisfies (*). (‘onverscly, suppose G satisfies (*) but for some subgroups CJ and I,- of G, 1.. is a maximal, but nonnormal, subgroup of V;. Then for some g E V, 1;~ !& L’. Thus 1 -C LT. l.S’,’ < V, so that :Cir CT”; : I’. But then R E ‘I-, L-0 and myc I-, contradicting (A). 1 Further properties of :T-groups will be investigated in the nest section. In particular it will be shown (Theorem 4.6) that an iv-group satisfies the strong basis property if and only if it is primary, thus determining completely those Brandt semigroups (not groups with zero) with the strong basis property-. TVe begin the proof of Theorem 3.1 by proving necessity, using the above Icmma.

I,mum 3.3.

If S is a Brandt semigroup which is not a group with zero, and

142

P. R. JOiYiES

S has the strong basis property, then any maximal iiT-group with the strong basis propert),.

nonxero subgroup of S is an

P~ooj. Let S be such a semigroup-- clearly each subgroup has the strong basis property. Suppose f~ E(S\O) IS . such that the maximal subgroup H, is not an lv-group, that is, H, does not satisfy (*). Then there exists y E Hi and L:,f 5:: H, such that y E
FIG. 1.

.I%“-classes of S.

Now

Also .X --- ay--I 6 (U, a> and a x xy E (U, x, y>, so (U, a> = (U, x,y>. But (U, y) c Ii, U H, u {0}, so a $ (U, y). The strong basis property for S gives a E (Ii, X) = (U, u Uf, X) = (xU,xpl u lYr , X) == (U, , x). Therefore y E (Ur , x>. But y $ U, , by assumption, so every expression for y as a product in (U, , x> must involve x, and further by Result 1.3(iv) cannot begin with .E

BASIS PROPERTIES FOR IXVERSE SEMIGROUPS

143

or end in X-I (since ~9%). By Corollary 1.4(i), y can be represented by an expression in (0; , CC)without idempotent factors. If such an expression begins with XPl, that is y = N QL for some u! E ( ZJ, , ~1 , 0) or with u E ZTf , since x- lUf ~ {Oj then ZL’cannot begin with 1vml(since 9 (by Result 1.3(iv)), a contradiction. Otherwise, the expression must begin with an element of ZJ, followed by x-1 (since Zl+ = {0)), and since the expression does not end with X-I, a contradiction is obtained in the same way. 1 Therefore every maximal nonzero subgroup is an Iv-group. Sufficiency in Theorem 3.1 will be proved in a sequence of three lemmas. First, some notation is required. Let S be a Brandt semigroup and s E S\,O. Put H(s) = H, u H,+ u H,,., U H,,-1, . Then H(s)O (=H(s) u (0)) is an inverse subsemigroup of S. If U < S, put U(s) = ( Z7 n H(s))O. Then C’(s) < H(s)O and U(s) < Go. Note also that if t E S and &;‘t2’z~, for some u, z’ E H(s), then t E H(s). LEMMA 3.4. Let S be a Brandt semigroup, U < S and s, y, a E S\,UO. If (U, x, y’>, = (U, a), then there exist x’, y’ E H(a) such that (C(a), x’, y’, -= ,:U(a), a’\, (U, x’) == (U, x\ and
Proof. Suppose l2 < S, x’, JP,a t S\ U” and (CT, x, 1,; = co’, a). Now since .y E (‘c- a’,l co I can be expressed in the form .X =- UX’V, where II, 2: E Z:r \,/’ > and x’ is a product (in (U, a::) beginning and ending in a or a~ l, so that (Corollary 1.4(ii)) x’ E H(a). Then X’ =_ us 1~~~-l, so ( Z’, .v; (CT, x’). Similarly y’ E H(a) can be found such that (c’, 3,~‘1:m: (Z:, y’,\. Therefore (Ii’, x’, y”, (U, n). Now x’ E (CT, a\\U. If X’ ==:UZC,where u E U and IC’begins with at, E := 2.1, then by Result 1.3(k) and (v), x’9?uJFa- f. so u E H(a). If s’ =-- z~ruz+ , where u E U, ZL’~ends with a’ and 2~‘~begins with a”, E, 7 L- 3.. I, then a-CW&‘a~ 7~, so u E N(a). Finally, if x’ = ZL’U,where u E G and u: ends in a<, E 2 & 1, then a--~&&x’ so u E H(a). So any factor ZI E Z in the product is in fact in Z’(a), that is i’ E (U(a), a). Similarly y’ E (Z:(a), a\ and N E (C’(a), x’, y’ , so (U(a), .x’, y’/\ = (U(a), a).

1

COROLLARY 3.5. Let S be a Brandt semiglroup.If ecevy incerse subsemlguoup H(s)O, s E S\O has the strong basis property, then so does S.

Proof. Let U < S, x, y, a E S and (U, X, y) = (U, a). We must show a E (U, x) or a E (U, y). If x, y or a is in U u {0}, this is immediate. So assume x, y, a E S\U”. From the previous lemma, there exist x’, y’ in H(a) such that (U, x) =- (U, x’), (U, y) = (U, y’) and (U(a), x’, y’) = (U(a), a) < H(a)O. Now if H(a)O has the strong basis property, then a E (U(a), x’), without loss of generality. So, since a # 0, we have a E (U, x’) = (U, x), as required. 1

144

P. K. JONES

LenInIA 3.6. Let S be a Brandt semigroup zchich is not a group with zero. If each maximal nonzero subgroup of S is an &--group with the strong basis property, then S has the strong basis property. Proof. By the previous corollary, it is suficient to show N(s)” has the strong basis property for any s t S 0. So let s t ,\” 0. If s is in a subgroup of S, then H(s)” :-. H,O, which has the strong basis property since FJ,?is a maximal subgroup. From now on, assume s is not in a subgroup. Let 1.- :< I](S)” and .Y,y, a E Il(s with ~,c’, s, My == / C, a . Again wc must show either a E (,‘, s or a E / 1., y;, and again, this is immediate if s, -V or a is in r,“‘. So we may assume X, y, a $ Go. E-l, and U(a) = (I; n M,) u Suppose a is in a subgroup of N(s); then H(a) {Ot, so from Lemma 3.4 there exist x’, y’ E II,, such that :.; tl,“, ;ljT, .v ,:U, y’, . Since II,” has ‘C’, .x’, and C’, y the strong basis propert!-, tither a E ,‘I,-(a), s’ or a c <
ul(arla

‘)u2 ... u,,(uu,,a -‘)

ul(g ‘(xz$-‘)~g)u2

“’ z&g

1(.Y2’,ts ‘)g)

E \ l’,. ,‘g ‘(xupY--yg (C‘,, ) .YL’,.Y l , a subgroup of If,. . Then ,y t L, I,“,, , so R EL, Put L since II, satisfies (*), by Lemma 3.2. Therefore a mapg I.Y F C’,. , .L.C’~.Z’~‘,s [ <~C:?, C?f, s, = cc-, s This completes

the proof of Theorem

4.

3.1.

N-GROUPS

In this section we consider groups with the strong basis property, and particularly, in view of Theorem 3.1, those which in addition are iv-groups. (A genera1 classification of all groups with the strong basis property is as yet

B.%SIS PROPERTIES

FOR ISVERSE

14.5

SEMGROUPS

not known.) This leads to Theorem 4.8, the final version of our answer to Ia. The first lemma shows that we are dealing with quite a restricted class of fpKlp’. L~n~lra 4.1. If the group G has the basis propert\,, prime-power order (for oarious primes).

theu ez,erJ’ element has

PFOC$ If a E G and has infinite order, then (a) and {a’, a3} are bases for contradicting the basis property. Thus G is .a of different cardinalities, periodic. I-urther if a has order mn, with (IN, n) :: 1, then {a> and {a”l, aThI are bases of different cardinalities. So every element has prime-power order. 1 Croups of this kind were first studied by Higman LKVRIA 4.1.

Ezery primary

[S].

i2beZian group has the strong basis property.

Proof. If G is a cyclic p-group, the subgroups of G form a chain, so G clearly has the strong basis property, by Result 1.1. If G is an arbitrary primary Abelian group, L :< G and s, J, a E G with C‘, a,,, then C4 G and in G/C:, ‘Ic’x, l-l,, (Za 3. Thus 1 .) .Y‘,J’ ~1C’a,) is a cyclic p-group and so I;;a E (1,:s or ( UJJ‘ . In G, therefore, a E
6.161.

Elements of relatively

prime, $nite

orde)

‘I’he E’rattini subgroup Q(G) (or l+(G)) of a group G is [I I] the intersection of all its maximal subgroups, or if G has no maximal subgroups, the group itself. Equivalently, Q(G) is th e set of nongenerators of G (where an element (76 G is said to be a nongenerator of G if (A, g) ~~7G implies (/I> = G for am ‘zubset .-I of G). It is a normal subgroup of G, and @(G/@(G)) is trivial. . LEMMA 4.4. Denote by G’ the commutator G1 C Q(G) .for an-v iiT-group G.

subgroup of the group G. Then

Prooj. Let M be a maximal subgroup of G; then M 4 G, and since G/M has no proper subgroups, it is a cyclic group of prime order. Hence G/M is =\belian and G’ CM. Therefore G1 C a(G). 1

146

P. R. JONFS

PROPOSITION; 4.5. .-I group G has the strong basis property ij- arId on/~* lf H,‘@(H) has the strolz,? basis property for every subgroup H of G. Z%ooj. Kccessit!- is immediate from Corollary 2.7. Conversely, suppose I’ 5, G, m, y, a E G and <.?Y,X, F; I, a; 11, Ski!'. Put di = Q(H). In HI@, we have (LT@jsP, x@,_v@:: == i CT@;@,a@ , and since Hi@ has the strong basis property- we may assume without loss of generalit!that a@ E ‘~:LV/@‘, .A@ =- s 1.‘, .T~@/d). Thus a = wg for some 7:~E
The following

group is nilpotent)

are equivalent fey an lv-group

(i)

G has the basis proput?;

(ii)

G is prinlarJ!;

(iii)

G has the strong basis propeTiT.

G:

Proof. (i) =- (ii). If G h as the basis property, every element of G has finite, prime-power order, by Lemma 4.1. But if there exist elements of relativel! prime order they commute by Lemma 4.3, and so their product does not ha\-e prime-power order. So every element has order a power of the same prime. (ii) --z (iii). Suppose G is primary, and let H zg G. ‘Then II,‘@(H) is an LT-group with trivial Frattini subgroup, and thus with trivial commutator subgroup (Lemma 4.4) that is H/@(H) is a primary abelian group. Ry Lemma 4.2, H/@(H) has the strong basis property. Hence G has the strong basis property, by Proposition 4.5. (iii) =zO(i) is trivial.

1

Since every finite p-group is nilpotent and thus an x-group, we have the following. (So Theorem 4.6 is a generalization of the Burnside Basis Theorem [II, 7.3.101). COROLLARY 4.7. Every jkite hence the basis property). 1

p-group

has the strong basis property

(and

Kot every p-group is an x-group, as Robinson [lo] illustrates with the example of infinite, finitely generated simple p-groups (which have maximal, nonnormal subgroups). However, locally finite p-groups are locally nilpotent and conversely, so in view of the remarks preceding Lemma 4.3, every locally finite p-group is a primary Iv-group. At this point we may also ask if there exist infinite p-groups without the [strong] basis property.

147

BASIS PROPERTIES FOR ISVERSE SEMIGROUP~

‘l’llcc)rcnls 1.3, 3.1, and 4.6 can novv be combined to giv-c our main theorem. ‘1’0 distinguisll those subgroups of an inverse semigroup whose associated principal factors are groups, or groups with xro, we will call an idempotcnt P, and any subgroup of II,: , isolated if e. ‘f for any idempotent f distinct from e, ? . that is, if IT, is an entire ‘i’-class. Othcrvvtse, call c, and each subgroup of HC , 7roll;.Folciift/.

?'IIEOREM 4.8 (Extended Basis Theorem). .ctroncgbasis propertQ8 if and only if:

mrd

,477 incense sem@oup

(i)

,V is completely semisimple;

II

ever>’ nonisolated maximal subgroup of S is a primary

S has the

K-group:

(“I (iii)

rze,y isolated maximal subgroup has the strong basis property.

Proof. Combine Result I .2 and Theorems comments preceding Corollary 2.4. 1 Some special cases may be quickly

2.3, 3.1 and 4.6, noting

the

noted.

4.0. if S is a completely semisimple incerse semi,oroup in which eseq* maximal subgroup is a primaq~ fi-group, in particular (i) if S is a finite inverse sem&oup in zuhich each subgroup is primary, OY (ii) if S is complete~\~ semisimple and X 7 L (for example, free inzjerse semzjyoups), the71 S has the strong basis property, and hence the basis property. COHOI.IAHY

Proof. The general result follows from the main theorem and Theorem 4.6, the case (i) from the fact that every finite inverse semigroup is completely semisimple [3], and (ii) trivially. 1 ‘The case (ii) is just the Basis Theorem

5. FINITF

(Corollary

2.4) of [9].

GROL~PS

In the last section we saw that the general problem of finding which groups have (either of) the basis properties is rather complex. The more powerful more machinery available for finite groups, however, enables considerably to be said. I:or instance, solubility can be proved fairly easily (although using some deep results). For definitions and properties of finite groups, the reader is referred to [7]. \Ve begin with a lemma similar to Proposition 4.5. Recall from Section 1 that an inverse semigroup S (in particular a group) is said to have the generating prnperf~* if an! two bases for S have the same cardinalitv.

148

3’. H. JONES

LENSI. 5. I. Let G be a jnitely generated group with the genes-atiq pyopert),. Then Gjk- has the generating property for easy novmal subgroup K of G contained in Q(G). I'mof. Put @ Q(G). If K 1 1 G and h’ . CD, then there czist surjecti1.e homomorphisms G 1, G,‘K 2 G/D. Since G has the generatine propert!, each basis of G has r elements, for some v t IN. Suppose IT ) >‘1 . .\‘? ,._.l J’,,,I 12 , so li,y contains a hasi % is a basis for G/K, 111E N. ‘I’hen G,‘@ l.:? i, 3.‘-> 19,,,j of J- and k _ w. Put _ ‘Y and picli .xj t J*,,,f ‘, 1 . j k. Put Al. ,. 1 , s2 ,..., .xicj. Since \&) - i, I I ; ’ k. -Y is irredundant. f;urther, if :f is a (finite) basis for C. and (I i- .I. 2: ‘0 i: then I E -Y(&) , so ~(.fg) zc(iy) for some zc t .Y l’hus r!,, @, that is 0 .Y. 0’ \\ IlelK zc$,, E *\-, +(, ‘I’herefore C; ker(k) CD’ ;$A,, a E :I, ’ is a finite subset of @. Hv the “nongcrwatcx” propert\. of CD, G .Y , that is .Y is a basis for G. So .\ k I’. Hut -Vj‘ 1:. % is) and is thus a basis for G,‘A. contained in 1 is irredundant, (since I.‘,;r of r elements, I’ 111.so 111 1.. i ‘I’herefore ever\- basis for G/K has K elements. ~OROLIARI. 5.2. !f G is a finite group z’ith the basis p~‘opfrt)‘, fh(‘t! WNJ’ homomorphic image of G has the basis property.

Proof It is sufficient to show that if G is a finite group with the basis propcrt!., and K -5 G, then G,‘K has the generatin‘? property. If KC D(G), then G/K has the generating property by the previous lemma. h-31. Otherwise G has a masimal subgroup III such that Kg M, so that G h-.-l. Then K n .-I I @(.I). for othwC’hoose --1 : G minimal such that G G. (G is a wdzlce(I product wise .d has a maximal subgroup .d’ such that K.-l’ of A hy A.3: [2; p. 641). K.-l,K 1~~.-I:K n .-I, which has the generating property 1,) ‘I’hcreforc GiK the previous lemma. 1 :I miuimo/ simple group is a simple non-Xbelian subgroup is soluhk. I.,ExL~

5.1.

group such that vi-cl-!- proper

-4 finite minimal simple group cannot hace the basis propert?’

Proof. 13~ a thcorcm of Thompson [ 121, a finite minimal simple group C can be generated by (a basis of) two elements. Further, by the Fcit~-‘l’hompson theorem on the soluhility of groups of odd order [6], G has even order. ‘Thus G possesses an involution. Let -\’ bc the subgroup generated by the involutions of G. ‘l’hcn .\- is a non:\‘. Thus G has a basis I of involutions. trivial normal subgroup of G, so G Rut I : i.:: 3, since two involutions generate a dihedral group. ‘I’herefol-c i G does not ha\-e the basis property.

BASIS

THEOREM

5.4.

PROPERTIES

FOR INVERSE

SEMIGROU’S

AJ finite group with the basis property

149

is soluble.

Proof. Suppose G is a counterexample of least order. If LV is a nontrivial normal subgroup of G, then by C’orollary 5.2, GiAY has the basis property. ‘Therefore G Ilut then :I- and G;‘S arc soluble, JO that G is, a contradiction. is simple, and in fact minimal simple. contradicting Lemma 5.3. 1 H&man [S] has classified the soluble finite groups in which every element has prime-power order. For any such group G there is a unique prime p such that G has a nontrivial normal p-subgroup. Let P be the maximum normal F(G), the Fitting subgroup of G). Either (i) p-subgroup of G. (Thus I P G, that is, G is a p-group; (ii) G/P is a cyclic g-group, for some prime q -I~ p; (iii) G/P is a generalized quaternion group ( p ; 2); or (iv) j G,‘P j ~-= pfrqh (q ,’ p), where y - 1 p”. In anv case at most two primes divide the o&t of G. (A more elegant approach is to note that G is a CI.\-group, that is the centralizer of each nonidentity element is nilpotent [5], and use the known structure of soluble C.Y-groups and Frobenius groups (see [7, (‘hap. 141)). From the previous section n-e know all finite p-groups have the strong basis property. Dickson [4] has shown that groups of types (iii) and (iv) cannot have the basis propert)-. Further a group G of type (ii) with the strong basis ply (CY1 I). For groups of this type property has G/P q. Thus G the basis property and the strong basis propert!- are equivalent. However there exist groups of type (ii) with G.‘P ~ ‘z- q having the basis property (but not the strong basis propert!.). This answers (II) in the negative. (A more “semigroup-orientated” answer will be given in Section 6!) Dickson and the author have in fact determined completely the finite groups with the basis property or the strong basis property. The complexity of the results and their proofs puts them beyond the scope of this paper, but we hope to present them in a further paper. 11-e illustrate this discussion with two examples which will be used in the nezt section. (Their properties will, howc\-er, be found independently of the discussion). I~GUPIX.

dihedral

5.5. Each non-Abelian group of order pq, and in particular group I),, ( p. q primes) has the strong basis property.

each

I+ooJ. Since the subgroup lattice of a group of order pq has length 2, the result is immediate from Result I .l. 1 5.6. Let G be the semidirect product of K by H, where K = s , P, H <‘II‘\, ?7’ 7. e, and h -I.rlr : 9. Then G has the basis property but not the strong basis property. ESAI\IPI.E

,$

Z+OO$ Let C -- (I?/ < ff. Then (11,.vj is a l .-basis for G, since G = : Kff. Rut I~~(/~.~)~ h2h2(k1.~h).~ : .?, so “.v, = s3 S 1 -, h.v , and therefore

150

P. R. JOXES

h E <,C, h.v/ also. So fhs] is a C-basis for G, and G does not have the strong basis property. Now the proper subgroups are just K; I-i and its four conjugates, and their subgroups of order two; and D, the (dihedral) normal subgroup of order 10, generated by the involutions of G. Thus each proper subgroup of G has the generating property, and in addition, we see that any basis for G must contain an element v of order 4, and an element z $ ,~y;. But every subgroup of order 4 is maximal, and so “‘I, z G. Thus every basis for G has two elements, and G has the generating property. 1

6.

‘The analog to Theorem stated.

THE

BASIS

PROPERTY

3.1 for the basis property

is more straightforwardly

THEOREM 6.1. Let S be a Brandt sem&oup with maximal (nonzero) subgroup G. Then S has the basis property ;f and onl~v if G has.

Proof Suppose each maximal subgroup of S has the basis property. Let s E S. If x E E == E(S), then clearly (x) is the only basis for (.xi; if x is in a (s, s i, xx r, subgroup, any basis for \<.v,, has one element; otherwise ‘s 1 x i.x] u (0) and (xl and {.x-l] are the only bases for I,,E E’ and we can write s Since x0 E (I;:, :‘-I. Thus N,$‘J~,~. By C’orolla& 1.4(i), we ma! assume F,, & R, and, by t

BASIS

PROPERTIES

FOR

INVERSE

SEMIGROUPS

151

replacing yu by its inverse if necessary, that x,,~Y~, . Put Ir =- yOycl. We can now repeat the above procedure to obtain a set Y’ such that (y’, yo> = (y, ya), (y E I), I” CJ (R, UL,) = m, 1 Y’\)h 1 = m - 1 and Y’\)h is irredundant. FVe will show (X’\g\ = (Y’\h>, whence by the induction hypothesis, m - 1 =: II ~~ 1, that is m :z n. Let y’ E Y’\h _C(S’jg, x0>, Now if $’ appears in an expression for y’ in .lyg, X”), it must be followed by x&r, for xaz = 0 unless z E R, , and the expression cannot end in x0 (as y’ $L,). But then the idempotent factor .r,,x;’ can be deleted. Similarly if sc;’ appears in such an expression, it must be preceded by s0 and can again be deleted. Therefore y’ E
answer to (II).

EXAMPLE 6.2. The Brandt semigroup S == A!‘O(S, ; I, I; d) (where S, is the symmetric group on 3 letters, and I is any set) has the basis property but not the strong basis property.

I’Yoo~. That S does not have the strong basis property follows from ‘Theorem 4.8, since S, is not primary-. But Sa G Da has the basis property, by Example 5.5, so S does. 1 Xs a corollary to Theorem 2.3 it was shown that a semilattice of groups has the strong basis property if and only if each subgroup has. That the analogous statement for the basis property is not true is seen by the next example. Let G be a finite group with the basis property but without the strong basis property (Example 5.6). Then there exist a subgroup U of G and U-irredundant subsets {s, y} and {u} of G with (c’, CX’, y) --= (U, a>. Let H be a group isomorphic with L’. ~XVIPLE 6.3. Let S be the semilattice of groups H U G, with linking homomorphism [3, Theorem 4.111 the natural monomorphism 4: H - G. Then each subgroup of S has the basis property, but S has not.

Proof. Let B be a (finite) basis for H. Then B U {x, y} and B u (a} are irredundant, since JI, > Jo , b E B, ( so, for instance, b # ((B U {a})\b)) and {x, y} and {u} are C-irredundant. Further (B u {x, y}j =: (U, x, y> u ti m:1 , so S does not have the basis property.

REFERENCES

1. R. BAER, Nilpotent groups and their generalizations, Z’rans. Amer. Math. SOC.47 (1940), 393-434. 481/50/1-11

152

P. K. JOSES

2. H. BECHTEM., “Theory of Groups,” Addison-Wesley, Reading, Mass., 1971. 3. .4. H. CLIFFORD AND G. B. PRESTON, “Algebraic Theory of Semigroups,” Math.

Surveys 7, Amer. Mnth. Sot., Providence, R.I., Vol. I, 1961, Vol. II, 1967. 4. N. K. DICKSON, private correspondence. 5. W. FEIT, iV1. HALL JR. AND J. G. THOMPSON, Finite groups in xvhich the centralizer of any non-identity element is nilpotent. MnQz. 2. 74 (1960), l-17. 6. W. FEIT AND J. G. THO~IPSON, Solvability of groups of odd order, Pacific /. Math. 13 (1963), 775-1029. 7. D. GORWSTEIN, “Finite Groups,” Harper and Row, N.Y. 1968. 8. G. HICMAN, Finite groups in which every element has prime power order, J. Londo~z Math. Sac. 32 (1957), 335-342. 9. P. R. JONES, A basis theorem for free inverse semigroups, J. Algebra, 49 (1977), 172-l 90. 10. D. J. S. ROBINSON, “Finiteness Conditions and Generalized Soluble Groups, Part 2,” Springer-Verlag, Berlin, 1972. Prentice-Hall, New Jersey, 1964. II. ‘c$‘. SCOTT, “Group Theory,” 12. J. G. THOMPSON, Nonsolvable finite groups all of %-hose local subgroups are solvable, Bzrll. Amer. Math. Sot. 74 (1968). 383-437. 13. J. S. WILSON, On periodic generalized nilpotent groups, Bull. London Math. Sot. 9 (1977), 81-85.