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Units of Burnside Rings of Elementary Abelian 2-Groups
Journal of Algebra 237, 487᎐500 Ž2001. doi:10.1006rjabr.2000.8589, available online at http:rrwww.idealibrary.com on
Units of Burnside Rings of Elementary Abelian 2-Groups Michael A. Alawode Department of Mathematics, Uni¨ ersity of Ibadan, Ibadan, Nigeria Communicated by Walter Feit Received June 7, 1999
INTRODUCTION Let G be a finite group. Then the set SŽ G . of G-isomorphism classes of all finite Žleft. G-sets forms a semi-ring under addition and multiplication induced, respectively, by the disjoint union and cartesian product. The Grothendieck ring of SŽ G . is called the Burnside ring of G and is denoted by ⍀ Ž G .. Let ⍀ Ž G .* be the group of units of the Burnside ring of G. Let G be an elementary Abelian 2-group. In Section 1 of this paper, we study subgroups of the character group Char Ž G . s : G ª "1 4 < Ž a1 ⭈ a2 . s Ž a1 . Ž a2 . ᭙ a1 , a2 g G 4 and prove the following main result: If X ; Char Ž G .
and for
a1 , a2 , . . . , a k g G,
put X Ž a1 , a2 , . . . ; a k . [ g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 4 ; then the following are equivalent: Ža. 噛 g X < < U s 1 CharŽU . for all subgroups U F G with < U < F 2 k 4 ' 0 mod 2. Žb. 噛 g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 ᭙ a1 , . . . , a k g G4 ' 0 mod 2. Žc. 噛X Ž a1 , a2 , . . . , a k . ' 0 mod 2 for all a1 , a2 , . . . , a k g G with l Ž a i . F 1 for all i s 1, 2, . . . , k. 487 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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MICHAEL A. ALAWODE
In Section 2, we study ⍀ Ž G .* as a Burnside ring module. First we identify the group ŽCharŽ G .. Žformed by the power set of CharŽ G . under symmetric difference. with ⍀ Ž G .* as a subgroup of "14 SubŽG ., where SubŽ G . s conjugacy classes of subgroups. This is done through a map
: Ž Char Ž G . . ª ⍀ Ž G . * : "1 4
Sub Ž G .
given by Ž X . s M X , where M X : SubŽ G . ª "14 is given by M X Ž H . s Ž y1 . 噛
gX < Ž h .s1 ᭙ hgH 4
.
We then obtain a filtration of ⍀ Ž G .* for < G < s 2 n : ⍀ Ž G . * s ⍀y1 Ž G . * > ⍀ 0 Ž G . * > ⭈⭈⭈ > ⍀ n Ž G . *. 1. SUBGROUPS OF CHARACTER GROUP ŽCharŽ G .. OF ELEMENTARY ABELIAN 2-GROUPS G Let N [ 1, 2, 3, . . . 4 , N0 [ 0 4 j N,
n g N0 .
Let n
G [ "1 4 s Ž ⑀ 1 , . . . , ⑀ n . < ⑀ i g "1 4 4 be the elementary Abelian 2-group of order 2 n. Note that G is a group relative to componentwise multiplication with 1 G s Ž1, 1, . . . , 1. and G , ⺪ 2 = ⺪ 2 = ⭈⭈⭈ = ⺪ 2 .
^
Define a function
`
n times
_
l : G ª N0 by l Ž Ž ⑀ 1 , . . . , ⑀ n . . s 噛 i g 1, . . . , n4 < ⑀ i s y1 4 . Call l a length function on G. Let Char Ž G . [ : G ª "1 4 < Ž a1 ⭈ a2 . s Ž a1 . ⭈ Ž a2 . for all a1 , a2 g G 4 . Then CharŽ G . is a group relative to argumentwise multiplication with identity character 1 CharŽG . : G ª "1 4 ,
a ¬ q1.
UNITS OF BURNSIDE RINGS OF
2-GROUPS
489
The map Char Ž G . ª G
¬ Ž Ž e1 . , . . . , Ž e n . . with e i [ Ž 1, . . . , 1, y1, 1, . . . , 1 . ith position
for all i s 1, . . . , n is a group isomorphism. For X : CharŽ G . and a1 , . . . , a k g G for some k g N0 , put X Ž a1 , . . . , a k . [ g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 4 . LEMMA 1.1. has
For all a1 , . . . , a ky1 , b1 , b 2 g G and all X : CharŽ G . one
X Ž a1 , . . . , a ky1 , b1 b 2 . s X Ž a1 , . . . , a ky1 , b1 . ^ X Ž a1 , . . . , a ky1 , b 2 . ^ X Ž a1 , . . . , a ky1 . , where for arbitrary sets Y, Z one puts
˙ ŽZ y Y ., Y ^Z [ ŽY y Z. j noting that
Ž Y ^ Z1 . ^ Z 2 s Y ^ Ž Z1 ^ Z 2 . . Proof. Because X s Y ^Z m Y s Z^ X, it is enough to verify that X Ž a1 , . . . , a ky1 , b1 ⭈ b 2 . ^ X Ž a1 , . . . , a ky1 . s X Ž a1 , . . . , a ky1 , b1 . ^ X Ž a1 , . . . , a ky1 , b 2 . . But L.H.S.s g X Ž a1 , . . . , a ky1 . < Ž b1 ⭈ b 2 . s y1 4
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MICHAEL A. ALAWODE
and R.H.S.s g X Ž a1 , . . . , a ky1 . < Ž b1 . s 1 and Ž b 2 . s y1 or
Ž b1 . s y1 and Ž b 2 . s 1 4 ; hence, L.H.S.s R.H.S.
LEMMA 1.2.
For arbitrary subsets Y1 , Y2 , . . . , Yn : CharŽ G . one has
噛 Y ^ Y2 ^ ⭈⭈⭈ ^ Yn s
ž
/
Ý
Ž y2.
Ž 噛T .y1
/T: 1, 2, . . . , n4
⭈ 噛 Ž Fi g T Yi . .
Proof. We shall supply a proof of this by induction. First let us check the formula for two sets, say Y, Y2 Žsee Fig. 1.. We have 噛Y1 s 噛 Ž Y1rY2 . q 噛 Ž Y1 l Y2 . 噛Y2 s 噛 Ž Y2rY1 . q 噛 Ž Y1 l Y2 . 噛Y1 q 噛Y2 s 噛 Ž Y1rY2 . q 噛 Ž Y2rY1 . q 2噛 Ž Y1 l Y2 . 噛 Y1rY2 j Y2rY1 s 噛Y1 q 噛Y2 y 2噛 Ž Y1 l Y2 . ,
ž
/
and so we have 噛 Ž Y1 ^Y2 . s 噛Y1 q 噛Y2 y 2噛 Ž Y1 l Y2 . ; hence, the formula is true for n s 2.
FIGURE 1
2-GROUPS
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491
Next, let us assume that the formula holds for n y 1 sets; that is, 噛 Y1 ^ Y2 ^ ⭈⭈⭈ ^ Yny1 s
ž
/
Ž y2.
Ý
Ž 噛T .y1
/T: 1, 2, . . . , ny1 4
⭈ 噛 Ž Fi g T Yi . .
It follows from our previous result Žequivalent to the case n s 2. that 噛 Y1 ^ Y2 ^ ⭈⭈⭈ ^ Yny1 ^ Yn
ž
/
s 噛 Y1 ^ Y2 ^ ⭈⭈⭈ ^ Yny1 ^ Yn
ž
/
s 噛 Y1 ^ Y2 ^ ⭈⭈⭈ ^ Yny1 q 噛Yn y 2噛 Ž Y1 ^ ⭈⭈⭈ ^ Yny1 . l Yn
ž
/
s 噛 Ž Y1 ^ ⭈⭈⭈ ^ Yny1 . q 噛Yn y 2噛 Ž Y1 l Yn . ^ ⭈⭈⭈ ^ Yny1 l Yn ,
/
s
Ž y2.
Ý
Ž 噛T .y1
/T: 1, 2, . . . , ny1 4
⭈ 噛 Ž Fi g T Yi . q 噛Yn
y 2噛 Ž Y1 l Yn . ^ ⭈⭈⭈ ^ Ž Yny1 l Yn . . The first term can be rewritten as
Ž y2.
Ý
Ž 噛T .y1
/T: 1, 2, . . . , ny1, n4 , nfT
⭈ 噛 Ž Fi g T Yi .
and the second and third terms together can be rewritten as
Ž y2.
Ý
Ž 噛T .y1
/T: 1, 2, . . . , n4 , ngT
⭈ 噛 Ž Fi g T Yi . .
The above two results yield
Ý
Ž y2.
Ž 噛T .y1
/T: 1, 2, . . . , n4
⭈ 噛 Ž Fi g T Yi . ;
hence, the formula is true for all n, and therefore by the principle of induction we obtain 噛 Ž Y1 ^ ⭈⭈⭈ ^ Yn . s
Ý
Ž y2.
/T: 1, 2, . . . , n4
Ž 噛T .y1
⭈ 噛 Ž Fi g T Yi . .
An immediate application of the above formula is as follows. For n s 2, 噛 Ž Y1 ^ Y2 . ' 噛Y1 q 噛Y2
mod 2.
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MICHAEL A. ALAWODE
For n s 3, 噛 Y1 ^ Y2 ^ Y3 ' 噛Y1 q 噛Y2 q 噛Y3
ž
/
y 2 噛 Ž Y1 l Y2 . q 噛 Ž Y1 l Y3 . q 噛 Ž Y2 l Y3 .
mod 4.
For n sets, n
噛 Ž Y1 ^ ⭈⭈⭈ ^ Yn . '
Ý 噛Yi y 2 Ý is1
1Fi-jFn
噛 Yi l Yj
ž
/
mod 4.
As a consequence of the above analysis we have n
噛 Ž Y1 ^ ⭈⭈⭈ ^Yn . '
Ý 噛Yi
mod 2.
is1
Moreover, since 噛 Ž Y ^ Z . ' 噛Y q 噛Z
mod 2
for all sets Y, Z, Lemma 1.1 implies the following corollary: COROLLARY 1.2. has
For all X : CharŽ G . and a1 , . . . , a ky1 , b1 , b 2 g G one
噛X Ž a1 , . . . , a ky1 , b1 b 2 . ' 噛X Ž a1 , . . . , a ky1 , 1 G . q 噛X Ž a1 , . . . , a ky1 , b1 . q 噛X Ž a1 , . . . , a ky1 , b 2 . THEOREM 1.3.
mod 2.
If X : CharŽ G ., then the following are equi¨ alent.
Ža. 噛 g X < < U s 1 CharŽU . , for all subgroups U F G with < U < F 2 k 4 ' 0 mod 2. Žb. 噛 g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1, for all a1 , . . . , a k g G4 ' 0 mod 2. Žc. 噛X Ž a1 , . . . , a k . ' 0 mod 2, for all a1 , . . . , a k g G with l Ž a i . F 1 for all i s 1, . . . , k. Proof. Since a subgroup U F G of G can be generated by k elements a1 , . . . , a k from G if and only if < U < F 2 k , Ža. m Žb.. It is also clear that Žb. « Žc..
UNITS OF BURNSIDE RINGS OF
2-GROUPS
493
To show that Žc. « Žb. one may proceed by induction relative to l Ž a1 . q ⭈⭈⭈ ql Ž a k . . If l Ž a1 . q ⭈⭈⭈ ql Ž a k . s 0, then l Ž a1 . s ⭈⭈⭈ s l Ž a k . s 0 and therefore the claim 噛 g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 4 ' 0
mod 2
follows directly from our assumption. Now assume that our claim is true whenever l Ž a1 . q ⭈⭈⭈ ql Ž a k . F n
for some n g N
and assume that l Ž a1 . q ⭈⭈⭈ ql Ž a k . s n q 1
for some a1 , . . . , a k g G.
If l Ž a i . F 1 for all i s 1, . . . , k, our assumption implies directly that 噛 g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 4 ' 0
mod 2.
Otherwise l Ž a i . G 1 for some i g 1, . . . , k 4 , say, i s k, so that a k s b1 ⭈ b 2
for some b1 , b 2 g G, with l Ž b1 . , l Ž b 2 . - l Ž a k .
and therefore ky1
Ý l Ž a i . q l Ž bj . F n
for j s 1, 2.
is1
Hence 噛X Ž a1 , . . . , a ky1 , b1 . ' 0
mod 2,
噛X Ž a1 , . . . , a ky1 , b 2 . ' 0
mod 2,
噛X Ž a1 , . . . , a ky1 , 1 G . ' 0
mod 2,
as well as
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MICHAEL A. ALAWODE
by our induction hypothesis, and therefore 噛X Ž a1 , . . . , a ky1 , a k . s 噛X Ž a1 , . . . , a ky1 , b1 b 2 . ' 噛X Ž a1 , . . . , a ky1 , b1 . q 噛X Ž a1 , . . . , a ky1 , b 2 . q 噛X Ž a1 , . . . , a ky1 , 1 G . '0
mod 2 as claimed.
2. THE UNIT GROUPS OF BURNSIDE RINGS OF ELEMENTARY ABELIAN 2-GROUPS G AS BURNSIDE RING MODULES Let G be an elementary Abelian 2-group and let
Ž Char Ž G . . [ X N X : Char Ž G . 4 be the power set of CharŽ G .. It can be shown that ŽCharŽ G .. is a commutative finite groupᎏmore precisely an elementary Abelian 2-group under the symmetric difference ^ as group multiplication. For every subset X : CharŽ G ., define M X : Sub Ž G . ª "1 4 by H ¬ Ž y1 . 噛
gX < Ž h .s1,for all hgH 4
,
where SubŽ G . denotes the set of subgroups of G. Let A Ž G . * s M X N X : Char Ž G . 4 . Then AŽ G .* is a subgroup of the Žmultiplicative. group of all maps from SubŽ G . into "14 . For all X, Y : CharŽ G . we have MX ⭈ MY s MX ^ Y and M Y ⭈ M Y s M [ 1 AŽG .* 4 .
2-GROUPS
UNITS OF BURNSIDE RINGS OF
495
Moreover, AŽ G .* can be identified with ⍀ Ž G . * : "1 4
Sub Ž G .
;
we henceforth make this identification so that ⍀ Ž G . * s M X N X : Char Ž G . 4 . THEOREM 2.1. The map
Ž Char Ž G . . ª ⍀ Ž G . * defined by X ª MX is an isomorphism! Proof. First, we note that the map
Ž Char Ž G . . ª ⍀ Ž G . * is a well-defined homomorphism. n Second, ŽCharŽ G .. and ⍀ Ž G .* are both of the same order, 2 2 , since
and moreover by standard results of Matsuda w20x < ⍀ Ž G . *< s 2 2 . n
We now prove injectivity as follows. We know that M X s 1 ⍀ ŽG .*
if and only if X s .
M X s 1 ⍀ ŽG .* implies that M X Ž H . s 1 for all subgroups H of G. M X Ž H . s 1 if and only if 噛 g X < Ž h . s 1, for all h g H 4 ' 0 and if and only if X s .
mod 2
496
MICHAEL A. ALAWODE
We assume that 噛 g X < Ž h . s 1, for all h g H 4 ' 0
mod 2
to show first that the trivial character is not in X ! Let 1 be the trivial character. We choose for this case the subgroup H [ G. By definition of a trivial character we have for an arbitrary character that Ž h. s 1 for all h g G if and only if s 1. Hence,
g X < Ž h . s 1, for all h g G 4 s X l 1 4 and therefore 噛 g X < Ž h . s 1, for all h g G 4 s 噛Ž X l 14 . s
½
1,
if 1 g X
0,
if 1 f X .
It follows that
1 f X , and so the trivial character is not in X. Finally, we must show that no non-trivial character is in such an X ! Let be a non-trivial character. For such a non-trivial character consider H [ g g G < Ž g . s 14 , a subgroup of G. For any other non-trivial character ⬘, ⬘Ž g . s 1 for all g g H if and only if s ⬘. Hence,
⬘ g X < ⬘ Ž g . s 1, for all
g g H4 s X l 4
and therefore 噛 ⬘ g X < ⬘ Ž g . s 1, for all g g H 4 s 噛Ž X l 4 . s
½
1,
for g X
0,
if f X
UNITS OF BURNSIDE RINGS OF
2-GROUPS
497
and so we have
f X. Hence, no non-trivial character is in such an X; therefore, X s , as claimed. Surjectivity follows from all the above considerations. Thus,
Ž Char Ž G . . ª ⍀ Ž G . * is an isomorphism. Put ⍀ k Ž G . * [ M X g ⍀ Ž G . * < M X Ž H . s 1, for all H F G with < H < F 2 k 4 , so that ⍀ Ž G . * s ⍀y1 Ž G . * > ⍀ 0 Ž G . * > ⍀ 1 Ž G . * > ⭈⭈⭈ > ⍀ n Ž G . * [ 1 ⍀ ŽG .* 4 s M 4 . LEMMA 2.2. ⍀ k Ž G .* s M X g ⍀ ky1Ž G .* < for all subsets T of 1, 2, . . . ,n4 of cardinality k; the number of g X with Ž e i . s 1 for all i g T is e¨ en4 . LEMMA 2.3.
⍀ k Ž G . * s ker
T : ⍀ ky1
Ł 1, 2, . . . , n 4 Tg k
ž
ž Ž G . * ª "14
/
1, 2, . . . , n 4 k
/
0
,
where
T : ⍀ ky1 Ž G . * ª "1 4 is the homomorphism which maps e¨ ery M X g ⍀ ky1 Ž G . *
onto M X Ž ² e i N i g T : . .
Proof. Given that T : 1, 2, . . . , n4 with 噛T s k, consider for each such T the map
T : ⍀ Ž G . * ª "1 4 : M X ¬ Ž y1 . 噛
gX < Ž e i .s1 for all igT 4
We contend that T is a homomorphism!
.
498
MICHAEL A. ALAWODE
For M X , M X ⬘ g ⍀ Ž G .*, we obtain
T Ž M X . [ M X Ž ² e i N i g T : . [ Ž y1 . 噛
gX < Ž e i .s1 for all igT 4
T Ž M X ⬘ . [ M X Ž ² e i N i g T : . [ Ž y1 . 噛
gX ⬘ < Ž e i .s1 for all igT 4
T Ž M X . ⭈ T Ž M X ⬘ . s Ž y1 . 噛 s Ž y1 . 噛
, ,
gX < Ž e i .s1 ᭙ igT 4q噛 gX ⬘ < Ž e i .s1 ᭙ igT 4 gX ^ X ⬘ < Ž e i .s1 ᭙ igT 4
,
since 噛 Ž X ^ X ⬘ . ' 噛X q 噛X ⬘
mod 2.
Hence, by definition, we have
Ž y1. 噛 gX ^ X ⬘ < Ž e i .s1 ᭙ igT 4 s MX ^ X ⬘ Ž² ei N i g T :. s MX ⭈ MX ⬘ Ž² ei N i g T :. s T Ž M X ⭈ M X . and therefore
T Ž M X ⬘ ⭈ M X . s T Ž M X . ⭈ T Ž M X ⬘ . as claimed. Now, since the number of sets of T : 1, 2, . . . , n4 with 噛T s k is nk , we shall have nk homomorphisms of such T . Thus
ž/
ž/
T : ⍀ ky1
Ł Tg
ž
1, 2, . . . , n 4 k
ž Ž G . * ª "14
1, 2 , . . . , n 4 k
/
/
is also a homomorphism. Therefore by 2.1 and the construction of ⍀ k Ž G .* above, we conclude that
⍀ k Ž G . * s ker
T : ⍀ ky1
Ł 1, 2, . . . , n 4 Tg k
ž
ž Ž G . * ª "14
/
Theorem 2.4.
Ž ⍀ ky 1Ž G . *:
n
⍀ k Ž G . *. s 2 Ž k . .
1, 2, . . . , n 4 k
/
0
.
UNITS OF BURNSIDE RINGS OF
2-GROUPS
499
APPENDIX: NOMENCLATURE Throughout this paper we use the following notations:
y1.
G is an elementary Abelian 2-group. 噛X or < X < is the cardinal number of a set X. 1 ⍀ ŽG . is the unit element wpointx of ⍀ Ž G .. R* is the unit group of a ring R. ⺪ is the ring of rational integers. ⺪ 2 [ "14 is a set having q1 and y1 as its elements. e i s Ž1, . . . , 1, y1, 1, . . . , 1. is an element of G, where the ith entry is
ž
1 , 2 , . . . , n 4 k
/ is the set of all subsets of order k of the set A s 1, 2, . . . , n4
where k, n are fixed positive integers, k F n. SubŽ G . is the subgroup lattice of G. l: G ª N0 is a length function on G 0 , where N0 [ 04 j N is the set of natural numbers N in disjoint union with the singleton set 04 , having 0 as its only element. ^ is the symmetric difference. ACKNOWLEDGMENTS I am sincerely grateful to my Ph.D. thesis supervisors Prof. Dr. Andreas Dress and Professor Aderemi O. Kuku for their guidance, patience, and generosity.
REFERENCES 1. M. A. Alawode, The Group of Units of Burnside Rings of Various Finite Groups, Ph.D. thesis, University of Ibadan, Ibadan, Nigeria, 1999. 2. R. Araki, Equivalent stable homotopy theory and idempotents of Burnside rings, Publ. Res. Inst. Math. Sci. 18 Ž1982., 1193᎐1212. 3. H. Bender, On groups with abelian Sylow 2-subgroups, Math. Z. 117 Ž1970., 164᎐176. 4. C. W. Curtis and I. Reiner, ‘‘Methods of Representation Theory,’’ Vols. 1 and 2, Wiley᎐Interscience, New York, 1981. 5. T. Dieck, ‘‘Transformation Groups and Representation Theory,’’ Lecture Notes in Mathematics, Vol. 766, Springer-Verlag, BerlinrNew York, 1979. 6. A. Dress, A characterization of solvable groups, Math. Z. 110 Ž1969., 213᎐217. 7. A. Dress, Operations in representation rings, in ‘‘Pro Symposia in Pure Mathematics,’’ pp. 39᎐45, 1971. 8. A. Dress, Contributions to the theory of induced representations, in ‘‘Algebraic K-Theory II, Proceedings of the Battle Institute Conference, 1972,’’ Lecture Notes in Mathematics, Vol. 342, pp. 183᎐240, Springer-Verlag, BerlinrNew York, 1973.
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9. A. Dress, Notes on the theory of representations of finite groups, Bielefeld Notes Ž1971.. 10. A. Dress and M. Kuchler, Zur Darstellungstheorie endlicher Gruppen I, Bielefeld Notes Ž1970.. 11. W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 Ž1963., 775᎐1029. 12. D. Gluck, Idempotent formula for the Burnside algebra with applications to the P-subgroup simplicial complex, Illinois J. Math. 25 Ž1981., 63᎐67. 13. D. Gorenstein, ‘‘Finite Groups,’’ Harper & Row, New York, 1968. 14. J. A. Green, Axiomatic representation theory for finite groups, J. Pure Appl. Algebra 1 Ž1971., 41᎐77. 15. W. H. Greub, ‘‘Multilinear Algebra,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1967. 16. W. H. Gustafson, Burnside rings which are Gorenstein, Comm. Algebra 5 Ž1977., 1᎐16. 17. R. Keown, ‘‘An Introduction to Group Representation Theory,’’ 1975. 18. A. O. Kuku, Axiomatic theory of induced representation of finite groups, in ‘‘Group Representation and Its Applications: Les Cours du C.I.M.P.A.’’ ŽA. O. Kuku, Ed.., 1985. 19. I. Li, Burnside algebra of a finite inverse semigroup, Zap. Nauchn. Steklo¨ . Inst. 46 Ž1974., 41᎐52; J. So¨ iet Math. 9 Ž1978., 322᎐331. 20. T. Matsuda, On the unit groups of Burnside rings, Japan. J. Math. Ž N.S.. 8 Ž1982., 71᎐93. 21. T. Matsuda, A note on the unit groups of the Burnside rings as Burnside ring modules, J. Fac. Sci. Shinshu Uni¨ . 21Ž1. Ž1986.. 22. T. Matsuda and T. Miyata, On the unit groups of the Burnside rings of finite groups, J. Math. Soc. Japan 35 Ž1983., 345᎐354. 23. H. Sasaki, Green correspondence and transfer theorems of Wielandt type for G-functors, J. Algebra 79 Ž1982., 98᎐120. 24. J. P. Serre, ‘‘Linear Representation of Finite Groups,’’ Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, BerlinrHeidelbergrNew York. 25. J. H. Walter, Finite groups with abelian Sylow 2-subgroups, Ann. of Math. 89 Ž1969., 405᎐514. 26. T. Yoshida, Character-theoretic transfer, J. Algebra 52 Ž1978., 1᎐38. 27. T. Yoshida, On G-functors. I. Transfer theorems for cohomological G-functors, Hokkaido Math. J. 9 Ž1980., 222᎐257. 28. T. Yoshida, Idempotents of Burnside rings and Dress induction theorem, J. Algebra 80 Ž1983., 90᎐105. 29. T. Yoshida, Idempotents and transfer theorems of Burnside rings, character rings and span rings, in ‘‘Algebraic and Topological Theories,’’ pp. 589᎐615, Kinokuniya, Tokyo, 1985. 30. T. Yoshida, On the unit groups for Burnside rings, J. Math. Soc. Japan 42Ž1. Ž1990..
Units of Burnside Rings of Elementary Abelian 2-Groups Michael A. Alawode Department of Mathematics, Uni¨ ersity of Ibadan, Ibadan, Nigeria Communicated by Walter Feit Received June 7, 1999
INTRODUCTION Let G be a finite group. Then the set SŽ G . of G-isomorphism classes of all finite Žleft. G-sets forms a semi-ring under addition and multiplication induced, respectively, by the disjoint union and cartesian product. The Grothendieck ring of SŽ G . is called the Burnside ring of G and is denoted by ⍀ Ž G .. Let ⍀ Ž G .* be the group of units of the Burnside ring of G. Let G be an elementary Abelian 2-group. In Section 1 of this paper, we study subgroups of the character group Char Ž G . s : G ª "1 4 < Ž a1 ⭈ a2 . s Ž a1 . Ž a2 . ᭙ a1 , a2 g G 4 and prove the following main result: If X ; Char Ž G .
and for
a1 , a2 , . . . , a k g G,
put X Ž a1 , a2 , . . . ; a k . [ g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 4 ; then the following are equivalent: Ža. 噛 g X < < U s 1 CharŽU . for all subgroups U F G with < U < F 2 k 4 ' 0 mod 2. Žb. 噛 g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 ᭙ a1 , . . . , a k g G4 ' 0 mod 2. Žc. 噛X Ž a1 , a2 , . . . , a k . ' 0 mod 2 for all a1 , a2 , . . . , a k g G with l Ž a i . F 1 for all i s 1, 2, . . . , k. 487 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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MICHAEL A. ALAWODE
In Section 2, we study ⍀ Ž G .* as a Burnside ring module. First we identify the group ŽCharŽ G .. Žformed by the power set of CharŽ G . under symmetric difference. with ⍀ Ž G .* as a subgroup of "14 SubŽG ., where SubŽ G . s conjugacy classes of subgroups. This is done through a map
: Ž Char Ž G . . ª ⍀ Ž G . * : "1 4
Sub Ž G .
given by Ž X . s M X , where M X : SubŽ G . ª "14 is given by M X Ž H . s Ž y1 . 噛
gX < Ž h .s1 ᭙ hgH 4
.
We then obtain a filtration of ⍀ Ž G .* for < G < s 2 n : ⍀ Ž G . * s ⍀y1 Ž G . * > ⍀ 0 Ž G . * > ⭈⭈⭈ > ⍀ n Ž G . *. 1. SUBGROUPS OF CHARACTER GROUP ŽCharŽ G .. OF ELEMENTARY ABELIAN 2-GROUPS G Let N [ 1, 2, 3, . . . 4 , N0 [ 0 4 j N,
n g N0 .
Let n
G [ "1 4 s Ž ⑀ 1 , . . . , ⑀ n . < ⑀ i g "1 4 4 be the elementary Abelian 2-group of order 2 n. Note that G is a group relative to componentwise multiplication with 1 G s Ž1, 1, . . . , 1. and G , ⺪ 2 = ⺪ 2 = ⭈⭈⭈ = ⺪ 2 .
^
Define a function
`
n times
_
l : G ª N0 by l Ž Ž ⑀ 1 , . . . , ⑀ n . . s 噛 i g 1, . . . , n4 < ⑀ i s y1 4 . Call l a length function on G. Let Char Ž G . [ : G ª "1 4 < Ž a1 ⭈ a2 . s Ž a1 . ⭈ Ž a2 . for all a1 , a2 g G 4 . Then CharŽ G . is a group relative to argumentwise multiplication with identity character 1 CharŽG . : G ª "1 4 ,
a ¬ q1.
UNITS OF BURNSIDE RINGS OF
2-GROUPS
489
The map Char Ž G . ª G
¬ Ž Ž e1 . , . . . , Ž e n . . with e i [ Ž 1, . . . , 1, y1, 1, . . . , 1 . ith position
for all i s 1, . . . , n is a group isomorphism. For X : CharŽ G . and a1 , . . . , a k g G for some k g N0 , put X Ž a1 , . . . , a k . [ g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 4 . LEMMA 1.1. has
For all a1 , . . . , a ky1 , b1 , b 2 g G and all X : CharŽ G . one
X Ž a1 , . . . , a ky1 , b1 b 2 . s X Ž a1 , . . . , a ky1 , b1 . ^ X Ž a1 , . . . , a ky1 , b 2 . ^ X Ž a1 , . . . , a ky1 . , where for arbitrary sets Y, Z one puts
˙ ŽZ y Y ., Y ^Z [ ŽY y Z. j noting that
Ž Y ^ Z1 . ^ Z 2 s Y ^ Ž Z1 ^ Z 2 . . Proof. Because X s Y ^Z m Y s Z^ X, it is enough to verify that X Ž a1 , . . . , a ky1 , b1 ⭈ b 2 . ^ X Ž a1 , . . . , a ky1 . s X Ž a1 , . . . , a ky1 , b1 . ^ X Ž a1 , . . . , a ky1 , b 2 . . But L.H.S.s g X Ž a1 , . . . , a ky1 . < Ž b1 ⭈ b 2 . s y1 4
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MICHAEL A. ALAWODE
and R.H.S.s g X Ž a1 , . . . , a ky1 . < Ž b1 . s 1 and Ž b 2 . s y1 or
Ž b1 . s y1 and Ž b 2 . s 1 4 ; hence, L.H.S.s R.H.S.
LEMMA 1.2.
For arbitrary subsets Y1 , Y2 , . . . , Yn : CharŽ G . one has
噛 Y ^ Y2 ^ ⭈⭈⭈ ^ Yn s
ž
/
Ý
Ž y2.
Ž 噛T .y1
/T: 1, 2, . . . , n4
⭈ 噛 Ž Fi g T Yi . .
Proof. We shall supply a proof of this by induction. First let us check the formula for two sets, say Y, Y2 Žsee Fig. 1.. We have 噛Y1 s 噛 Ž Y1rY2 . q 噛 Ž Y1 l Y2 . 噛Y2 s 噛 Ž Y2rY1 . q 噛 Ž Y1 l Y2 . 噛Y1 q 噛Y2 s 噛 Ž Y1rY2 . q 噛 Ž Y2rY1 . q 2噛 Ž Y1 l Y2 . 噛 Y1rY2 j Y2rY1 s 噛Y1 q 噛Y2 y 2噛 Ž Y1 l Y2 . ,
ž
/
and so we have 噛 Ž Y1 ^Y2 . s 噛Y1 q 噛Y2 y 2噛 Ž Y1 l Y2 . ; hence, the formula is true for n s 2.
FIGURE 1
2-GROUPS
UNITS OF BURNSIDE RINGS OF
491
Next, let us assume that the formula holds for n y 1 sets; that is, 噛 Y1 ^ Y2 ^ ⭈⭈⭈ ^ Yny1 s
ž
/
Ž y2.
Ý
Ž 噛T .y1
/T: 1, 2, . . . , ny1 4
⭈ 噛 Ž Fi g T Yi . .
It follows from our previous result Žequivalent to the case n s 2. that 噛 Y1 ^ Y2 ^ ⭈⭈⭈ ^ Yny1 ^ Yn
ž
/
s 噛 Y1 ^ Y2 ^ ⭈⭈⭈ ^ Yny1 ^ Yn
ž
/
s 噛 Y1 ^ Y2 ^ ⭈⭈⭈ ^ Yny1 q 噛Yn y 2噛 Ž Y1 ^ ⭈⭈⭈ ^ Yny1 . l Yn
ž
/
s 噛 Ž Y1 ^ ⭈⭈⭈ ^ Yny1 . q 噛Yn y 2噛 Ž Y1 l Yn . ^ ⭈⭈⭈ ^ Yny1 l Yn ,
/
s
Ž y2.
Ý
Ž 噛T .y1
/T: 1, 2, . . . , ny1 4
⭈ 噛 Ž Fi g T Yi . q 噛Yn
y 2噛 Ž Y1 l Yn . ^ ⭈⭈⭈ ^ Ž Yny1 l Yn . . The first term can be rewritten as
Ž y2.
Ý
Ž 噛T .y1
/T: 1, 2, . . . , ny1, n4 , nfT
⭈ 噛 Ž Fi g T Yi .
and the second and third terms together can be rewritten as
Ž y2.
Ý
Ž 噛T .y1
/T: 1, 2, . . . , n4 , ngT
⭈ 噛 Ž Fi g T Yi . .
The above two results yield
Ý
Ž y2.
Ž 噛T .y1
/T: 1, 2, . . . , n4
⭈ 噛 Ž Fi g T Yi . ;
hence, the formula is true for all n, and therefore by the principle of induction we obtain 噛 Ž Y1 ^ ⭈⭈⭈ ^ Yn . s
Ý
Ž y2.
/T: 1, 2, . . . , n4
Ž 噛T .y1
⭈ 噛 Ž Fi g T Yi . .
An immediate application of the above formula is as follows. For n s 2, 噛 Ž Y1 ^ Y2 . ' 噛Y1 q 噛Y2
mod 2.
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MICHAEL A. ALAWODE
For n s 3, 噛 Y1 ^ Y2 ^ Y3 ' 噛Y1 q 噛Y2 q 噛Y3
ž
/
y 2 噛 Ž Y1 l Y2 . q 噛 Ž Y1 l Y3 . q 噛 Ž Y2 l Y3 .
mod 4.
For n sets, n
噛 Ž Y1 ^ ⭈⭈⭈ ^ Yn . '
Ý 噛Yi y 2 Ý is1
1Fi-jFn
噛 Yi l Yj
ž
/
mod 4.
As a consequence of the above analysis we have n
噛 Ž Y1 ^ ⭈⭈⭈ ^Yn . '
Ý 噛Yi
mod 2.
is1
Moreover, since 噛 Ž Y ^ Z . ' 噛Y q 噛Z
mod 2
for all sets Y, Z, Lemma 1.1 implies the following corollary: COROLLARY 1.2. has
For all X : CharŽ G . and a1 , . . . , a ky1 , b1 , b 2 g G one
噛X Ž a1 , . . . , a ky1 , b1 b 2 . ' 噛X Ž a1 , . . . , a ky1 , 1 G . q 噛X Ž a1 , . . . , a ky1 , b1 . q 噛X Ž a1 , . . . , a ky1 , b 2 . THEOREM 1.3.
mod 2.
If X : CharŽ G ., then the following are equi¨ alent.
Ža. 噛 g X < < U s 1 CharŽU . , for all subgroups U F G with < U < F 2 k 4 ' 0 mod 2. Žb. 噛 g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1, for all a1 , . . . , a k g G4 ' 0 mod 2. Žc. 噛X Ž a1 , . . . , a k . ' 0 mod 2, for all a1 , . . . , a k g G with l Ž a i . F 1 for all i s 1, . . . , k. Proof. Since a subgroup U F G of G can be generated by k elements a1 , . . . , a k from G if and only if < U < F 2 k , Ža. m Žb.. It is also clear that Žb. « Žc..
UNITS OF BURNSIDE RINGS OF
2-GROUPS
493
To show that Žc. « Žb. one may proceed by induction relative to l Ž a1 . q ⭈⭈⭈ ql Ž a k . . If l Ž a1 . q ⭈⭈⭈ ql Ž a k . s 0, then l Ž a1 . s ⭈⭈⭈ s l Ž a k . s 0 and therefore the claim 噛 g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 4 ' 0
mod 2
follows directly from our assumption. Now assume that our claim is true whenever l Ž a1 . q ⭈⭈⭈ ql Ž a k . F n
for some n g N
and assume that l Ž a1 . q ⭈⭈⭈ ql Ž a k . s n q 1
for some a1 , . . . , a k g G.
If l Ž a i . F 1 for all i s 1, . . . , k, our assumption implies directly that 噛 g X < Ž a1 . s ⭈⭈⭈ s Ž a k . s 1 4 ' 0
mod 2.
Otherwise l Ž a i . G 1 for some i g 1, . . . , k 4 , say, i s k, so that a k s b1 ⭈ b 2
for some b1 , b 2 g G, with l Ž b1 . , l Ž b 2 . - l Ž a k .
and therefore ky1
Ý l Ž a i . q l Ž bj . F n
for j s 1, 2.
is1
Hence 噛X Ž a1 , . . . , a ky1 , b1 . ' 0
mod 2,
噛X Ž a1 , . . . , a ky1 , b 2 . ' 0
mod 2,
噛X Ž a1 , . . . , a ky1 , 1 G . ' 0
mod 2,
as well as
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MICHAEL A. ALAWODE
by our induction hypothesis, and therefore 噛X Ž a1 , . . . , a ky1 , a k . s 噛X Ž a1 , . . . , a ky1 , b1 b 2 . ' 噛X Ž a1 , . . . , a ky1 , b1 . q 噛X Ž a1 , . . . , a ky1 , b 2 . q 噛X Ž a1 , . . . , a ky1 , 1 G . '0
mod 2 as claimed.
2. THE UNIT GROUPS OF BURNSIDE RINGS OF ELEMENTARY ABELIAN 2-GROUPS G AS BURNSIDE RING MODULES Let G be an elementary Abelian 2-group and let
Ž Char Ž G . . [ X N X : Char Ž G . 4 be the power set of CharŽ G .. It can be shown that ŽCharŽ G .. is a commutative finite groupᎏmore precisely an elementary Abelian 2-group under the symmetric difference ^ as group multiplication. For every subset X : CharŽ G ., define M X : Sub Ž G . ª "1 4 by H ¬ Ž y1 . 噛
gX < Ž h .s1,for all hgH 4
,
where SubŽ G . denotes the set of subgroups of G. Let A Ž G . * s M X N X : Char Ž G . 4 . Then AŽ G .* is a subgroup of the Žmultiplicative. group of all maps from SubŽ G . into "14 . For all X, Y : CharŽ G . we have MX ⭈ MY s MX ^ Y and M Y ⭈ M Y s M [ 1 AŽG .* 4 .
2-GROUPS
UNITS OF BURNSIDE RINGS OF
495
Moreover, AŽ G .* can be identified with ⍀ Ž G . * : "1 4
Sub Ž G .
;
we henceforth make this identification so that ⍀ Ž G . * s M X N X : Char Ž G . 4 . THEOREM 2.1. The map
Ž Char Ž G . . ª ⍀ Ž G . * defined by X ª MX is an isomorphism! Proof. First, we note that the map
Ž Char Ž G . . ª ⍀ Ž G . * is a well-defined homomorphism. n Second, ŽCharŽ G .. and ⍀ Ž G .* are both of the same order, 2 2 , since
and moreover by standard results of Matsuda w20x < ⍀ Ž G . *< s 2 2 . n
We now prove injectivity as follows. We know that M X s 1 ⍀ ŽG .*
if and only if X s .
M X s 1 ⍀ ŽG .* implies that M X Ž H . s 1 for all subgroups H of G. M X Ž H . s 1 if and only if 噛 g X < Ž h . s 1, for all h g H 4 ' 0 and if and only if X s .
mod 2
496
MICHAEL A. ALAWODE
We assume that 噛 g X < Ž h . s 1, for all h g H 4 ' 0
mod 2
to show first that the trivial character is not in X ! Let 1 be the trivial character. We choose for this case the subgroup H [ G. By definition of a trivial character we have for an arbitrary character that Ž h. s 1 for all h g G if and only if s 1. Hence,
g X < Ž h . s 1, for all h g G 4 s X l 1 4 and therefore 噛 g X < Ž h . s 1, for all h g G 4 s 噛Ž X l 14 . s
½
1,
if 1 g X
0,
if 1 f X .
It follows that
1 f X , and so the trivial character is not in X. Finally, we must show that no non-trivial character is in such an X ! Let be a non-trivial character. For such a non-trivial character consider H [ g g G < Ž g . s 14 , a subgroup of G. For any other non-trivial character ⬘, ⬘Ž g . s 1 for all g g H if and only if s ⬘. Hence,
⬘ g X < ⬘ Ž g . s 1, for all
g g H4 s X l 4
and therefore 噛 ⬘ g X < ⬘ Ž g . s 1, for all g g H 4 s 噛Ž X l 4 . s
½
1,
for g X
0,
if f X
UNITS OF BURNSIDE RINGS OF
2-GROUPS
497
and so we have
f X. Hence, no non-trivial character is in such an X; therefore, X s , as claimed. Surjectivity follows from all the above considerations. Thus,
Ž Char Ž G . . ª ⍀ Ž G . * is an isomorphism. Put ⍀ k Ž G . * [ M X g ⍀ Ž G . * < M X Ž H . s 1, for all H F G with < H < F 2 k 4 , so that ⍀ Ž G . * s ⍀y1 Ž G . * > ⍀ 0 Ž G . * > ⍀ 1 Ž G . * > ⭈⭈⭈ > ⍀ n Ž G . * [ 1 ⍀ ŽG .* 4 s M 4 . LEMMA 2.2. ⍀ k Ž G .* s M X g ⍀ ky1Ž G .* < for all subsets T of 1, 2, . . . ,n4 of cardinality k; the number of g X with Ž e i . s 1 for all i g T is e¨ en4 . LEMMA 2.3.
⍀ k Ž G . * s ker
T : ⍀ ky1
Ł 1, 2, . . . , n 4 Tg k
ž
ž Ž G . * ª "14
/
1, 2, . . . , n 4 k
/
0
,
where
T : ⍀ ky1 Ž G . * ª "1 4 is the homomorphism which maps e¨ ery M X g ⍀ ky1 Ž G . *
onto M X Ž ² e i N i g T : . .
Proof. Given that T : 1, 2, . . . , n4 with 噛T s k, consider for each such T the map
T : ⍀ Ž G . * ª "1 4 : M X ¬ Ž y1 . 噛
gX < Ž e i .s1 for all igT 4
We contend that T is a homomorphism!
.
498
MICHAEL A. ALAWODE
For M X , M X ⬘ g ⍀ Ž G .*, we obtain
T Ž M X . [ M X Ž ² e i N i g T : . [ Ž y1 . 噛
gX < Ž e i .s1 for all igT 4
T Ž M X ⬘ . [ M X Ž ² e i N i g T : . [ Ž y1 . 噛
gX ⬘ < Ž e i .s1 for all igT 4
T Ž M X . ⭈ T Ž M X ⬘ . s Ž y1 . 噛 s Ž y1 . 噛
, ,
gX < Ž e i .s1 ᭙ igT 4q噛 gX ⬘ < Ž e i .s1 ᭙ igT 4 gX ^ X ⬘ < Ž e i .s1 ᭙ igT 4
,
since 噛 Ž X ^ X ⬘ . ' 噛X q 噛X ⬘
mod 2.
Hence, by definition, we have
Ž y1. 噛 gX ^ X ⬘ < Ž e i .s1 ᭙ igT 4 s MX ^ X ⬘ Ž² ei N i g T :. s MX ⭈ MX ⬘ Ž² ei N i g T :. s T Ž M X ⭈ M X . and therefore
T Ž M X ⬘ ⭈ M X . s T Ž M X . ⭈ T Ž M X ⬘ . as claimed. Now, since the number of sets of T : 1, 2, . . . , n4 with 噛T s k is nk , we shall have nk homomorphisms of such T . Thus
ž/
ž/
T : ⍀ ky1
Ł Tg
ž
1, 2, . . . , n 4 k
ž Ž G . * ª "14
1, 2 , . . . , n 4 k
/
/
is also a homomorphism. Therefore by 2.1 and the construction of ⍀ k Ž G .* above, we conclude that
⍀ k Ž G . * s ker
T : ⍀ ky1
Ł 1, 2, . . . , n 4 Tg k
ž
ž Ž G . * ª "14
/
Theorem 2.4.
Ž ⍀ ky 1Ž G . *:
n
⍀ k Ž G . *. s 2 Ž k . .
1, 2, . . . , n 4 k
/
0
.
UNITS OF BURNSIDE RINGS OF
2-GROUPS
499
APPENDIX: NOMENCLATURE Throughout this paper we use the following notations:
y1.
G is an elementary Abelian 2-group. 噛X or < X < is the cardinal number of a set X. 1 ⍀ ŽG . is the unit element wpointx of ⍀ Ž G .. R* is the unit group of a ring R. ⺪ is the ring of rational integers. ⺪ 2 [ "14 is a set having q1 and y1 as its elements. e i s Ž1, . . . , 1, y1, 1, . . . , 1. is an element of G, where the ith entry is
ž
1 , 2 , . . . , n 4 k
/ is the set of all subsets of order k of the set A s 1, 2, . . . , n4
where k, n are fixed positive integers, k F n. SubŽ G . is the subgroup lattice of G. l: G ª N0 is a length function on G 0 , where N0 [ 04 j N is the set of natural numbers N in disjoint union with the singleton set 04 , having 0 as its only element. ^ is the symmetric difference. ACKNOWLEDGMENTS I am sincerely grateful to my Ph.D. thesis supervisors Prof. Dr. Andreas Dress and Professor Aderemi O. Kuku for their guidance, patience, and generosity.
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