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Enumeration of Metacyclicp-Groups
JOURNAL OF ALGEBRA ARTICLE NO.
186, 436]446 Ž1996.
0381
Enumeration of Metacyclic p-Groups Steven Liedahl Department of Mathematics, Uni¨ ersity of Notre Dame, Notre Dame, Indiana 46556-5683 Communicated by Walter Feit Received August 1, 1995
1. PRELIMINARIES A finite group G is called metacyclic if G has a cyclic normal subgroup N is cyclic. In this paper we use the N for which the quotient GrN parametrization of metacyclic p-groups given in w1x to determine the number of metacyclic p-groups of order p N, for any prime p. The two principal results, which are proved in Sections 2 and 3, are the following: THEOREM 3. of order p N is
If p is an odd prime, then the number of metacyclic p-groups
Ž N 3 q 12 N 2 q 12 N q 72. r72
if N ' 0 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 56. r72
if N ' 1 Ž mod 6 .
Ž N q 12 N q 12 N q 64. r72
if N ' 2 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 72. r72
if N ' 3 Ž mod 6 .
Ž N q 12 N q 12 N q 56. r72
if N ' 4 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 64. r72
if N ' 5 Ž mod 6 . .
3
3
2
2
There are 4 metacyclic groups of order 8, which include the dihedral and quaternion groups. THEOREM 7. 2 N is
If N G 4, then the number of metacyclic 2-groups of order
Ž N 3 q 48 N 2 y 168 N q 432. r72
if N ' 0 Ž mod 6 .
Ž N q 48 N y 177N q 416. r72
if N ' 1 Ž mod 6 .
Ž N 3 q 48 N 2 y 168 N q 424. r72
if N ' 2 Ž mod 6 .
3
2
436 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
ENUMERATION OF METACYCLIC
p-GROUPS
437
Ž N 3 q 48 N 2 y 177N q 432. r72
if N ' 3 Ž mod 6 .
Ž N 3 q 48 N 2 y 168 N q 416. r72
if N ' 4 Ž mod 6 .
Ž N q 48 N y 177N q 424. r72
if N ' 5 Ž mod 6 . .
3
2
The metacyclic p-groups were first enumerated by Lindenberg in w2x. A subsequent enumeration was mentioned in the paper w3x of Newman and Xu, but their work is unpublished. Lindenberg’s results slightly differ from ours. We find that for N G 5 the number of metacyclic groups of order 2 N is less than the number given in w2, Corrected Versionx by N y 4. ŽSee Theorem 6 and the remark which follows.. There are similarities between our methods and those of w2x. Both use roughly the same classification into three types, and both distinguish between split and nonsplit groups within the first two types. In the present work, however, the treatment of these cases benefits from parametrizations which immediately lead to summation formulas. Also the parametrizations used here are based on the simpler and self-contained classification given in w1x. We begin with a review of some results of w1x. A metacyclic p-group of order p N may be given by a presentation ² x, y < x p s 1, y p s x p , yxyy1 s x q : , n
m
s
where n q m s N, q p ' 1 Žmod p n ., and p n < p s Ž q y 1., and each such presentation defines a metacyclic group of order p N. The metacyclic p-groups naturally fall into three classes: m
Case 1.
p odd.
Case 2.
p s 2 and q ' 1 Žmod 4..
Case 3.
p s 2, q ' 3 Žmod 4., and G nonabelian. t
t
For groups in Cases 1, 2, and 3, we may assume q s Ž p q 1. p , q s 5 2 , t and q s y5 2 , respectively. A metacyclic group is called split if it is expressible as a semidirect product of two cyclic subgroups. For example, in Case 3 all dihedral 2-groups ² x, y < x 2 s 1, y 2 s 1, yxyy1 s xy1 : n
Ž n G 2.
are split, and all quaternion groups ² x, y < x 2 s 1, y 2 s x 2 n
ny 1
, yxyy1 s xy1 :
Ž n G 2.
are nonsplit. In Cases 1 and 2 it is natural to count separately the split and nonsplit groups. Lemmas 1, 2, and 3 below give a parametrization of the
438
STEVEN LIEDAHL
groups in Cases 1, 2, and 3, respectively. We use w x x to denote the greatest integer less than or equal to x. LEMMA 1. Let p be an odd prime. If N G 3, then each nonabelian split group of order p N is obtained exactly once by letting the parameters n, t, and s ¨ ary according to n g 2, 3, . . . , N y 1 4 t g max Ž 0, 2 n y N y 1 . , . . . , n y 2 4 s s min Ž n, N y n . . If N G 6, then each nonsplit group of order p N is obtained exactly once by letting the parameters n, t, and s ¨ ary according to n g 3, 4, . . . , Ž 2 N y 3 . r3
4
t g max Ž 0, 2 n y N . , . . . , min Ž n y 3, N y n y 3 . 4 s g max Ž t q 2, n y t y 1 . , . . . , min Ž n, N y n . y 1 4 . Proof. In the split case it is clear that n g 2, 3, . . . , N y 14 . The m inequality t G maxŽ0, 2 n y N . is implied by the congruence q p ' 1 Žmod p n ., while t F n y 2 follows from the isomorphism ŽZrp n Z.* ( C py 1 = C p ny 1 and the fact that p q 1 generates the factor of order p ny 1. The normalizations given in w1, Sect. 2.1x show that, for odd p, any split extension of cyclic p-groups is equivalent to one with s s minŽ n, N y n.. Conversely, if N G 3 then each triple Ž n, t, s . as above defines a distinct nonabelian split group of order p N by w1, Lemma 8x. The restrictions on n and t in the split case also apply in the nonsplit case. Any extension of cyclic p-groups, for odd p, is equivalent to one with s G n y t y 1 by w1, Lemma 2x. By w1, Remark 2.4.3x the inequality t q 1 - s - minŽ n, N y n. must be satisfied in the nonsplit case. This and the assumption that the sets above are nonempty imply that the stated restrictions on Ž n, t, s . are necessary. Conversely, for each N G 6, each such triple Ž n, t, s . defines a distinct nonsplit group of order p N by w1, Theorem 9, Remark 2.4.3x. The proof of Lemma 2 is similar and will be omitted. LEMMA 2. If N G 4, then each nonabelian split group of order 2 N in Case 2 is obtained exactly once by letting the parameters n, t, and s ¨ ary according to n g 3, 4, . . . , N y 1 4 t g max Ž 0, 2 n y N y 2 . , . . . , n y 3 4 s s min Ž n, N y n . .
ENUMERATION OF METACYCLIC
p-GROUPS
439
If N G 8, then each nonsplit group of order 2 N in Case 2 is obtained exactly once by letting the parameters n, t, and s ¨ ary according to n g 4, 5, . . . , Ž 2 N y 3 . r3
4
t g max Ž 0, 2 n y N y 1 . , . . . , min Ž n y 4, N y n y 4 . 4 s g max Ž t q 3, n y t y 2 . , . . . , min Ž n, N y n . y 1 4 . LEMMA 3. If N G 3, then each group of order 2 N in Case 3 is obtained by letting the parameters n, t, and s ¨ ary according to n g 2, 3, . . . , N y 1 4 t g max Ž 0, 2 n y N y 2 . , . . . , n y 2 4 s g n y 1, min Ž n, N y n q t q 1 . 4 . The resulting groups are distinct after omitting those for which N G 5, 3 F n F N y 2, t s n y 2, and s s n y 1. Proof. Clearly n g 2, 3, . . . , N y 14 must be satisfied. In Case 3 we t take q s y5 2 , and the inequality t G maxŽ0, 2 n y N y 2. follows m from q 2 ' 1 Žmod 2 n ., while t F n y 2 follows from the isomorphism ŽZr2 n Z.* ( C2 = C2 ny 2 and the fact that 5 generates the factor of order 2 ny 2 . We may assume s g n y 1, minŽ n, N y n q t q 1.4 by w1, Lemma 2x. Conversely, for N G 3 each triple Ž n, t, s . as above defines a group of order 2 N. We obtain distinct isomorphism classes after omitting the groups above according to w1, Theorem 22x. Finally, by w1, Lemma 23x, for each N G 4 there is a single 2-group which belongs to Cases 2 and 3. It has presentations ² x, y < x 2 s 1, y 2 s x 2 n
ny 1
, yxyy1 s x 1q2
ny 1
:
and ² x, y < x 4 s 1, y 2
ny 1
s x 2 , yxyy1 s xy1 : .
We will use the identities k
Ý j s k Ž k q 1. r2
Ž 1.
js1 k
Ý j 2 s Ž 2 k 3 q 3k 2 q k . r6. js1
Ž 2.
440
STEVEN LIEDAHL
Equation Ž1. implies n
Ý min Ž t , n q 1 y t . s ts1
½
Ž n2 q 2 n . r4, Ž n2 q 2 n q 1 . r4,
n ' 0 Ž mod 2 . n ' 1 Ž mod 2 . .
Ž 3.
2. THE CASE p ODD We begin with an enumeration of the nonabelian split groups. The smallest has order p 3 with parameters n s 2, t s 0, s s 1, and presentation ² x, y < x p s 1, y p s x p , yxyy1 s x pq1 : . 2
THEOREM 1. Let p be an odd prime. If N G 3 and F Ž N . denotes the number of nonabelian split metacyclic p-groups of order p N, then
½
FŽ N . s
Ž N 2 y 2 N . r4, Ž N 2 y 2 N q 1 . r4,
N ' 0 Ž mod 2 . N ' 1 Ž mod 2 . .
Proof. According to Lemma 1 we may write Ny1
FŽ N . s
card max Ž 0, 2 n y N y 1 . , . . . , n y 2 4 .
Ý ns2
Then maxŽ0, 2 n y N y 1. s 0 if and only if n F wŽ N q 1.r2x. Therefore wŽ Nq1 .r2 x
FŽ N . s
Ý
Ny1
Ž n y 1. q
Ý
Ž N y n. .
ns wŽ Nq3 .r2 x
ns2
Equation Ž1. then gives the result. We now turn to the nonsplit groups. The smallest has order p 6 with parameters n s 3, t s 0, s s 2, and presentation ² x, y < x p s 1, y p s x p , yxyy1 s x pq1 : . 3
3
2
ENUMERATION OF METACYCLIC
p-GROUPS
441
THEOREM 2. Let p be an odd prime. If N G 6 and GŽ N . denotes the number of nonsplit metacyclic p-groups of order p N, then
¡Ž N
3
y 6 N 2 q 12 N . r72,
N ' 0 Ž mod 6 .
Ž N y 6 N q 3 N q 2 . r72, Ž N 3 y 6 N 2 q 12 N y 8 . r72, G Ž N . s~ Ž N 3 y 6 N 2 q 3 N q 18. r72, Ž N 3 y 6 N 2 q 12 N y 16. r72, Ž N 3 y 6 N 2 q 3 N q 10. r72, 3
2
¢
N ' 1 Ž mod 6 . N ' 2 Ž mod 6 . N ' 3 Ž mod 6 . N ' 4 Ž mod 6 . N ' 5 Ž mod 6 . .
Proof. By Lemma 1 we have wŽ2 Ny3 .r3 x min Ž ny3, Nyny3 .
GŽ N . s
card max Ž t q 2, n y t y 1 . , . . . ,
Ý
Ý
ns3
tsmax Ž0, 2 nyN .
min Ž n, N y n . y 1 4 . Certain of these maxima and minima are determined as follows: N ' 0 Ž mod 2 .
n F Nr2
n ) Nr2
max Ž 0, 2 n y N . min Ž n y 3, N y n y 3 . min Ž n, N y n .
0 ny3 n
2n y N Nyny3 Nyn
N ' 1 Ž mod 2 .
n F Ž N y 1 . r2
n ) Ž N y 1 . r2
max Ž 0, 2 n y N . min Ž n y 3, N y n y 3 . min Ž n, N y n .
0 ny3 n
2n y N Nyny3 Nyn
Assume that N ' 0 Žmod 6.. According to the above, we may write Nr2 ny3
GŽ N . s
Ý Ý Ž n y max Ž t q 2, n y t y 1. . ns3 ts0 Ž2 Ny3 .r3
q
Nyny3
Ý Ž N y n y max Ž t q 2, n y t y 1. .
Ý
ns Ž Nq2 .r2 ts2 nyN Ž Ny4 .r2
s
Ý ns1
n
Ž Ny6 .r6 3ny2
Ý min Ž t , n q 1 y t . q Ý
Ý
ts1
ts1
ns1
min Ž t , 3n y 1 y t . .
442
STEVEN LIEDAHL
Using Eq. Ž3., if N ' 0 Žmod 12. then Ž Ny4 .r4
GŽ N . s
Ý Ž2 n
Nr12 2
q n. q
ns1
Ý Ž 9n2 y 12 n q 4. ns1
Ž Ny12 .r12
q
Ý
Ž 9n2 y 3n .
ns1
s s
1 96 1 72
Ž N 3 y 3N 2 y 4N . q
1 288
Ž N 3 y 15N 2 q 60 N .
Ž N 3 y 6 N 2 q 12 N . .
In case N ' 6 Žmod 12. we have Ž Ny6 .r4
GŽ N . s
Ý Žn
Ž Ny2 .r4 2
q n. q
ns1
s
1 96
Ý
Ž Ny6 .r12
n q 2
ns1
Ž N 3 y 3 N 2 y 4 N q 12 . q
Ý
Ž 18 n2 y 15n q 4 .
ns1
1 288
Ž N 3 y 15N 2 q 60 N y 36 . .
The computations for N k 0 Žmod 6. are similar, and will be omitted. Remark. The construction of the polynomials in Theorem 2 is based on the assumption N G 6, though the formulas correctly give the value 0 for 0 F N F 5. Similarly, the construction in Theorem 1 is based on the assumption N G 3. THEOREM 3. of order p N is
If p is an odd prime, then the number of metacyclic p-groups
Ž N 3 q 12 N 2 q 12 N q 72. r72
if N ' 0 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 56. r72
if N ' 1 Ž mod 6 .
Ž N 3 q 12 N 2 q 12 N q 64. r72
if N ' 2 Ž mod 6 .
Ž N q 12 N q 3 N q 72. r72
if N ' 3 Ž mod 6 .
Ž N 3 q 12 N 2 q 12 N q 56. r72
if N ' 4 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 64. r72
if N ' 5 Ž mod 6 . .
3
2
Proof. The number of two-generator abelian p-groups of order p N is Ž N q 2.r2 if N ' 0 Žmod 2., and the number is Ž N q 1.r2 if N ' 1 Žmod 2.. To this we add the results of Theorems 1 and 2.
p-GROUPS
ENUMERATION OF METACYCLIC
443
3. THE CASE p s 2 The enumeration procedure for Case 2 is essentially the same as for the case p odd. We begin with the nonabelian split groups. The smallest has order 16 and belongs to Case 3 as well. Its Case 2 parameters are n s 3, t s 0, and s s 1. Its Case 3 parameters are n s 2, t s 0, and s s 1. The group has presentations ² x, y < x 8 s 1, y 2 s x 2 , yxyy1 s x 5 : , and ² x, y < x 4 s 1, y 4 s x 2 , yxyy1 s xy1 : . THEOREM 4. If N G 4 and F2 Ž N . denotes the number of nonabelian split metacyclic 2-groups of order 2 N in Case 2, then
½
F2 Ž N . s
Ž N 2 y 4 N q 4 . r4, Ž N 2 y 4 N q 3 . r4,
N ' 0 Ž mod 2 . N ' 1 Ž mod 2 . .
Proof. According to Lemma 2 we have Ny1
F2 Ž N . s
card max Ž 0, 2 n y N y 2 . , . . . , n y 3 4 .
Ý ns3
Then maxŽ0, 2 n y N y 2. s 0 if and only if n F wŽ N q 2.r2x. Therefore wŽ Nq2 .r2 x
F2 Ž N . s
Ý
Ny1
Ž n y 2. q
ns3
Ý
Ž N y n. ,
ns wŽ Nq4 .r2 x
and the result follows from Eq. Ž1.. The smallest nonsplit group in Case 2 has order 2 8 , with parameters n s 4, t s 0, s s 3, and presentation ² x, y < x 16 s 1, y 16 s x 8 , yxyy1 s x 5 : . THEOREM 5. Assume N G 8. Let G 2 Ž N . denote the number of nonsplit metacyclic 2-groups of order 2 N in Case 2, and let GŽ N . be as defined in Theorem 2. Then G2 Ž N . s
½
G Ž N . y Ž N y 4 . r2, G Ž N . y Ž N y 5 . r2,
N ' 0 Ž mod 2 . N ' 1 Ž mod 2 . .
444
STEVEN LIEDAHL
Proof. By Lemma 2 we have wŽ2 Ny3 .r3 x min Ž ny4, Nyny4 .
G2 Ž N . s
card max Ž t q 3, n y t y 2 . , . . . ,
Ý
Ý
ns4
tsmax Ž0, 2 nyNy1 .
min Ž n, N y n . y 1 4 wŽ2 Ny3 .r3 x min Ž ny3, Nyny3 .
s
card max Ž t q 2, n y t y 1 . , . . . ,
Ý
Ý
ns4
tsmax Ž1, 2 nyN .
min Ž n, N y n . y 1 4 . Then maxŽ0, 2 n y N . s 0 if and only if n F w Nr2x, therefore wŽ2 Ny3 .r3 x min Ž ny3, Nyny3 .
G2 Ž N . s
card max Ž t q 2, n y t y 1 . , . . . ,
Ý
Ý
ns3
tsmax Ž0, 2 nyN . w Nr2 x
min Ž n, N y n . y 1 4 y
Ý
min Ž n y 2, 1 .
ns3
s G Ž N . y w Nr2x q 2, and the result follows. Remark. The assumption N G 8 is used in constructing the polynomials in Theorem 5, though they correctly give the value 0 for 4 F N F 7. Similarly, the assumption N G 4 is used in constructing the polynomials in Theorem 4, though they correctly give the value 0 for 1 F N F 3. Finally we count the 2-groups in Case 3. THEOREM 6. Case 3, then
If H Ž N . denotes the number of groups of order 2 N in
HŽ N . s
½
2, Ž N 2 y 3 N q 6 . r2,
Ns3 N G 4.
Proof. By Lemma 3 we obtain each such group exactly once by letting the parameters n, t, and s vary according to n g 2, . . . , N y 1 4 , t g max Ž 0, 2 n y N y 2 . , . . . , n y 2 4 , s g n y 1, min Ž n, N y n q t q 1 . 4 ,
ENUMERATION OF METACYCLIC
p-GROUPS
445
then, if N G 5, omitting each group for which 3 F n F N y 2, t s n y 2, and s s n y 1. It follows that H Ž3. s 2 and H Ž4. s 5. For N G 5 we have Ny1
ny2
Ý
Ý
HŽ N . s
card n y 1, min Ž n, N y n q t q 1 . 4
ns2 tsmax Ž0, 2 nyNy2 .
y Ž N y 4. . Observe that cardŽ n y 1, minŽ n, N y n q t q 1.4 s 1 if and only if t s 2 n y N y 2, and maxŽ0, 2 n y N y 2. s 2 n y N y 2 if and only if n G wŽ N q 3.r2x. If N ' 0 Žmod 2. then Nr2 ny2
HŽ N . s
ns2 ts0
s
Ny1
Ý Ý 2q 1 4
Ý ns Ž Nq2 .r2
Ž N2 y 2N. q
1 4
ny2
ž
1q
Ý ts2 nyNy1
/
2 y Ž N y 4.
Ž N 2 y 4. y Ž N y 4. .
If N ' 1 Žmod 2. then Ž Nq1 .r2 ny2
HŽ N . s
Ý ns2
s
1 4
Ny1
Ý 2q ts0
Ž N 2 y 1. q
Ý ns Ž Nq3 .r2
1 4
ny2
ž
1q
/
2 y Ž N y 4.
Ý ts2 nyNy1
Ž N 2 y 2 N y 3. y Ž N y 4. .
This completes the proof. Remark. In Theorem 6 each group ² x, y < x 2 s 1, y 2 s x 2 , yxyy1 s my 1 x with n G 3, m G 2 is discounted, as the generating set Ž xy 2 , y . 2n 2m 2 ny 1 y1 y1q2 ny 1 satisfies the relations z s 1, w s z , wzw s z . This accounts for the discrepancy of N y 4 between our enumeration and that of w2, Sect. 4.2x. n
m
ny 1
y1 :
The following gives the number of metacyclic groups of order 2 N for N G 4. There are, of course, four metacyclic groups of order 8. THEOREM 7. 2 N is
If N G 4, then the number of metacyclic 2-groups of order
Ž N 3 q 48 N 2 y 168 N q 432. r72
if N ' 0 Ž mod 6 .
Ž N q 48 N y 177N q 416. r72
if N ' 1 Ž mod 6 .
Ž N 3 q 48 N 2 y 168 N q 424. r72
if N ' 2 Ž mod 6 .
Ž N 3 q 48 N 2 y 177N q 432. r72
if N ' 3 Ž mod 6 .
3
2
446
STEVEN LIEDAHL
Ž N 3 q 48 N 2 y 168 N q 416. r72
if N ' 4 Ž mod 6 .
Ž N 3 q 48 N 2 y 177N q 424. r72
if N ' 5 Ž mod 6 . .
Proof. The number of two-generator abelian 2-groups of order 2 N is Ž N q 2.r2 if N ' 0 Žmod 2., and the number is Ž N q 1.r2 if N ' 1 Žmod 2.. To this we add the results of Theorems 4, 5, and 6, then subtract 1 according to the comment which follows Lemma 3. Remark. We observe that the nonsplit groups alone account for the terms of degree 3 in Theorems 3 and 7. To conclude, we give the number of nonabelian metacyclic groups of order p N for N F 15. N ps2 p odd
3 2 1
4 5 2
5 9 4
6 15 7
7 22 10
8 32 15
9 43 20
10 57 27
11 72 34
12 91 44
13 110 53
14 134 66
15 158 78
ACKNOWLEDGMENT The author is grateful for a Research Award from the School of Natural Sciences, California State University, Fresno.
REFERENCES 1. S. Liedahl, Presentations of metacyclic p-groups with applications to K-admissibility questions, J. Algebra 169 Ž1994., 965]983. 2. W. Lindenberg, Struktur und Klassifizierung bizyklischer p-Gruppen, Ber. Gesellsch. Math. Daten¨ erarb. 40 Ž1971., 1]36. 3. M. Newman and M.-Y. Xu, Metacyclic groups of prime-power order, Ad¨ . in Math. (Beijing) 17 Ž1988., 106]107.
186, 436]446 Ž1996.
0381
Enumeration of Metacyclic p-Groups Steven Liedahl Department of Mathematics, Uni¨ ersity of Notre Dame, Notre Dame, Indiana 46556-5683 Communicated by Walter Feit Received August 1, 1995
1. PRELIMINARIES A finite group G is called metacyclic if G has a cyclic normal subgroup N is cyclic. In this paper we use the N for which the quotient GrN parametrization of metacyclic p-groups given in w1x to determine the number of metacyclic p-groups of order p N, for any prime p. The two principal results, which are proved in Sections 2 and 3, are the following: THEOREM 3. of order p N is
If p is an odd prime, then the number of metacyclic p-groups
Ž N 3 q 12 N 2 q 12 N q 72. r72
if N ' 0 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 56. r72
if N ' 1 Ž mod 6 .
Ž N q 12 N q 12 N q 64. r72
if N ' 2 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 72. r72
if N ' 3 Ž mod 6 .
Ž N q 12 N q 12 N q 56. r72
if N ' 4 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 64. r72
if N ' 5 Ž mod 6 . .
3
3
2
2
There are 4 metacyclic groups of order 8, which include the dihedral and quaternion groups. THEOREM 7. 2 N is
If N G 4, then the number of metacyclic 2-groups of order
Ž N 3 q 48 N 2 y 168 N q 432. r72
if N ' 0 Ž mod 6 .
Ž N q 48 N y 177N q 416. r72
if N ' 1 Ž mod 6 .
Ž N 3 q 48 N 2 y 168 N q 424. r72
if N ' 2 Ž mod 6 .
3
2
436 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
ENUMERATION OF METACYCLIC
p-GROUPS
437
Ž N 3 q 48 N 2 y 177N q 432. r72
if N ' 3 Ž mod 6 .
Ž N 3 q 48 N 2 y 168 N q 416. r72
if N ' 4 Ž mod 6 .
Ž N q 48 N y 177N q 424. r72
if N ' 5 Ž mod 6 . .
3
2
The metacyclic p-groups were first enumerated by Lindenberg in w2x. A subsequent enumeration was mentioned in the paper w3x of Newman and Xu, but their work is unpublished. Lindenberg’s results slightly differ from ours. We find that for N G 5 the number of metacyclic groups of order 2 N is less than the number given in w2, Corrected Versionx by N y 4. ŽSee Theorem 6 and the remark which follows.. There are similarities between our methods and those of w2x. Both use roughly the same classification into three types, and both distinguish between split and nonsplit groups within the first two types. In the present work, however, the treatment of these cases benefits from parametrizations which immediately lead to summation formulas. Also the parametrizations used here are based on the simpler and self-contained classification given in w1x. We begin with a review of some results of w1x. A metacyclic p-group of order p N may be given by a presentation ² x, y < x p s 1, y p s x p , yxyy1 s x q : , n
m
s
where n q m s N, q p ' 1 Žmod p n ., and p n < p s Ž q y 1., and each such presentation defines a metacyclic group of order p N. The metacyclic p-groups naturally fall into three classes: m
Case 1.
p odd.
Case 2.
p s 2 and q ' 1 Žmod 4..
Case 3.
p s 2, q ' 3 Žmod 4., and G nonabelian. t
t
For groups in Cases 1, 2, and 3, we may assume q s Ž p q 1. p , q s 5 2 , t and q s y5 2 , respectively. A metacyclic group is called split if it is expressible as a semidirect product of two cyclic subgroups. For example, in Case 3 all dihedral 2-groups ² x, y < x 2 s 1, y 2 s 1, yxyy1 s xy1 : n
Ž n G 2.
are split, and all quaternion groups ² x, y < x 2 s 1, y 2 s x 2 n
ny 1
, yxyy1 s xy1 :
Ž n G 2.
are nonsplit. In Cases 1 and 2 it is natural to count separately the split and nonsplit groups. Lemmas 1, 2, and 3 below give a parametrization of the
438
STEVEN LIEDAHL
groups in Cases 1, 2, and 3, respectively. We use w x x to denote the greatest integer less than or equal to x. LEMMA 1. Let p be an odd prime. If N G 3, then each nonabelian split group of order p N is obtained exactly once by letting the parameters n, t, and s ¨ ary according to n g 2, 3, . . . , N y 1 4 t g max Ž 0, 2 n y N y 1 . , . . . , n y 2 4 s s min Ž n, N y n . . If N G 6, then each nonsplit group of order p N is obtained exactly once by letting the parameters n, t, and s ¨ ary according to n g 3, 4, . . . , Ž 2 N y 3 . r3
4
t g max Ž 0, 2 n y N . , . . . , min Ž n y 3, N y n y 3 . 4 s g max Ž t q 2, n y t y 1 . , . . . , min Ž n, N y n . y 1 4 . Proof. In the split case it is clear that n g 2, 3, . . . , N y 14 . The m inequality t G maxŽ0, 2 n y N . is implied by the congruence q p ' 1 Žmod p n ., while t F n y 2 follows from the isomorphism ŽZrp n Z.* ( C py 1 = C p ny 1 and the fact that p q 1 generates the factor of order p ny 1. The normalizations given in w1, Sect. 2.1x show that, for odd p, any split extension of cyclic p-groups is equivalent to one with s s minŽ n, N y n.. Conversely, if N G 3 then each triple Ž n, t, s . as above defines a distinct nonabelian split group of order p N by w1, Lemma 8x. The restrictions on n and t in the split case also apply in the nonsplit case. Any extension of cyclic p-groups, for odd p, is equivalent to one with s G n y t y 1 by w1, Lemma 2x. By w1, Remark 2.4.3x the inequality t q 1 - s - minŽ n, N y n. must be satisfied in the nonsplit case. This and the assumption that the sets above are nonempty imply that the stated restrictions on Ž n, t, s . are necessary. Conversely, for each N G 6, each such triple Ž n, t, s . defines a distinct nonsplit group of order p N by w1, Theorem 9, Remark 2.4.3x. The proof of Lemma 2 is similar and will be omitted. LEMMA 2. If N G 4, then each nonabelian split group of order 2 N in Case 2 is obtained exactly once by letting the parameters n, t, and s ¨ ary according to n g 3, 4, . . . , N y 1 4 t g max Ž 0, 2 n y N y 2 . , . . . , n y 3 4 s s min Ž n, N y n . .
ENUMERATION OF METACYCLIC
p-GROUPS
439
If N G 8, then each nonsplit group of order 2 N in Case 2 is obtained exactly once by letting the parameters n, t, and s ¨ ary according to n g 4, 5, . . . , Ž 2 N y 3 . r3
4
t g max Ž 0, 2 n y N y 1 . , . . . , min Ž n y 4, N y n y 4 . 4 s g max Ž t q 3, n y t y 2 . , . . . , min Ž n, N y n . y 1 4 . LEMMA 3. If N G 3, then each group of order 2 N in Case 3 is obtained by letting the parameters n, t, and s ¨ ary according to n g 2, 3, . . . , N y 1 4 t g max Ž 0, 2 n y N y 2 . , . . . , n y 2 4 s g n y 1, min Ž n, N y n q t q 1 . 4 . The resulting groups are distinct after omitting those for which N G 5, 3 F n F N y 2, t s n y 2, and s s n y 1. Proof. Clearly n g 2, 3, . . . , N y 14 must be satisfied. In Case 3 we t take q s y5 2 , and the inequality t G maxŽ0, 2 n y N y 2. follows m from q 2 ' 1 Žmod 2 n ., while t F n y 2 follows from the isomorphism ŽZr2 n Z.* ( C2 = C2 ny 2 and the fact that 5 generates the factor of order 2 ny 2 . We may assume s g n y 1, minŽ n, N y n q t q 1.4 by w1, Lemma 2x. Conversely, for N G 3 each triple Ž n, t, s . as above defines a group of order 2 N. We obtain distinct isomorphism classes after omitting the groups above according to w1, Theorem 22x. Finally, by w1, Lemma 23x, for each N G 4 there is a single 2-group which belongs to Cases 2 and 3. It has presentations ² x, y < x 2 s 1, y 2 s x 2 n
ny 1
, yxyy1 s x 1q2
ny 1
:
and ² x, y < x 4 s 1, y 2
ny 1
s x 2 , yxyy1 s xy1 : .
We will use the identities k
Ý j s k Ž k q 1. r2
Ž 1.
js1 k
Ý j 2 s Ž 2 k 3 q 3k 2 q k . r6. js1
Ž 2.
440
STEVEN LIEDAHL
Equation Ž1. implies n
Ý min Ž t , n q 1 y t . s ts1
½
Ž n2 q 2 n . r4, Ž n2 q 2 n q 1 . r4,
n ' 0 Ž mod 2 . n ' 1 Ž mod 2 . .
Ž 3.
2. THE CASE p ODD We begin with an enumeration of the nonabelian split groups. The smallest has order p 3 with parameters n s 2, t s 0, s s 1, and presentation ² x, y < x p s 1, y p s x p , yxyy1 s x pq1 : . 2
THEOREM 1. Let p be an odd prime. If N G 3 and F Ž N . denotes the number of nonabelian split metacyclic p-groups of order p N, then
½
FŽ N . s
Ž N 2 y 2 N . r4, Ž N 2 y 2 N q 1 . r4,
N ' 0 Ž mod 2 . N ' 1 Ž mod 2 . .
Proof. According to Lemma 1 we may write Ny1
FŽ N . s
card max Ž 0, 2 n y N y 1 . , . . . , n y 2 4 .
Ý ns2
Then maxŽ0, 2 n y N y 1. s 0 if and only if n F wŽ N q 1.r2x. Therefore wŽ Nq1 .r2 x
FŽ N . s
Ý
Ny1
Ž n y 1. q
Ý
Ž N y n. .
ns wŽ Nq3 .r2 x
ns2
Equation Ž1. then gives the result. We now turn to the nonsplit groups. The smallest has order p 6 with parameters n s 3, t s 0, s s 2, and presentation ² x, y < x p s 1, y p s x p , yxyy1 s x pq1 : . 3
3
2
ENUMERATION OF METACYCLIC
p-GROUPS
441
THEOREM 2. Let p be an odd prime. If N G 6 and GŽ N . denotes the number of nonsplit metacyclic p-groups of order p N, then
¡Ž N
3
y 6 N 2 q 12 N . r72,
N ' 0 Ž mod 6 .
Ž N y 6 N q 3 N q 2 . r72, Ž N 3 y 6 N 2 q 12 N y 8 . r72, G Ž N . s~ Ž N 3 y 6 N 2 q 3 N q 18. r72, Ž N 3 y 6 N 2 q 12 N y 16. r72, Ž N 3 y 6 N 2 q 3 N q 10. r72, 3
2
¢
N ' 1 Ž mod 6 . N ' 2 Ž mod 6 . N ' 3 Ž mod 6 . N ' 4 Ž mod 6 . N ' 5 Ž mod 6 . .
Proof. By Lemma 1 we have wŽ2 Ny3 .r3 x min Ž ny3, Nyny3 .
GŽ N . s
card max Ž t q 2, n y t y 1 . , . . . ,
Ý
Ý
ns3
tsmax Ž0, 2 nyN .
min Ž n, N y n . y 1 4 . Certain of these maxima and minima are determined as follows: N ' 0 Ž mod 2 .
n F Nr2
n ) Nr2
max Ž 0, 2 n y N . min Ž n y 3, N y n y 3 . min Ž n, N y n .
0 ny3 n
2n y N Nyny3 Nyn
N ' 1 Ž mod 2 .
n F Ž N y 1 . r2
n ) Ž N y 1 . r2
max Ž 0, 2 n y N . min Ž n y 3, N y n y 3 . min Ž n, N y n .
0 ny3 n
2n y N Nyny3 Nyn
Assume that N ' 0 Žmod 6.. According to the above, we may write Nr2 ny3
GŽ N . s
Ý Ý Ž n y max Ž t q 2, n y t y 1. . ns3 ts0 Ž2 Ny3 .r3
q
Nyny3
Ý Ž N y n y max Ž t q 2, n y t y 1. .
Ý
ns Ž Nq2 .r2 ts2 nyN Ž Ny4 .r2
s
Ý ns1
n
Ž Ny6 .r6 3ny2
Ý min Ž t , n q 1 y t . q Ý
Ý
ts1
ts1
ns1
min Ž t , 3n y 1 y t . .
442
STEVEN LIEDAHL
Using Eq. Ž3., if N ' 0 Žmod 12. then Ž Ny4 .r4
GŽ N . s
Ý Ž2 n
Nr12 2
q n. q
ns1
Ý Ž 9n2 y 12 n q 4. ns1
Ž Ny12 .r12
q
Ý
Ž 9n2 y 3n .
ns1
s s
1 96 1 72
Ž N 3 y 3N 2 y 4N . q
1 288
Ž N 3 y 15N 2 q 60 N .
Ž N 3 y 6 N 2 q 12 N . .
In case N ' 6 Žmod 12. we have Ž Ny6 .r4
GŽ N . s
Ý Žn
Ž Ny2 .r4 2
q n. q
ns1
s
1 96
Ý
Ž Ny6 .r12
n q 2
ns1
Ž N 3 y 3 N 2 y 4 N q 12 . q
Ý
Ž 18 n2 y 15n q 4 .
ns1
1 288
Ž N 3 y 15N 2 q 60 N y 36 . .
The computations for N k 0 Žmod 6. are similar, and will be omitted. Remark. The construction of the polynomials in Theorem 2 is based on the assumption N G 6, though the formulas correctly give the value 0 for 0 F N F 5. Similarly, the construction in Theorem 1 is based on the assumption N G 3. THEOREM 3. of order p N is
If p is an odd prime, then the number of metacyclic p-groups
Ž N 3 q 12 N 2 q 12 N q 72. r72
if N ' 0 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 56. r72
if N ' 1 Ž mod 6 .
Ž N 3 q 12 N 2 q 12 N q 64. r72
if N ' 2 Ž mod 6 .
Ž N q 12 N q 3 N q 72. r72
if N ' 3 Ž mod 6 .
Ž N 3 q 12 N 2 q 12 N q 56. r72
if N ' 4 Ž mod 6 .
Ž N 3 q 12 N 2 q 3 N q 64. r72
if N ' 5 Ž mod 6 . .
3
2
Proof. The number of two-generator abelian p-groups of order p N is Ž N q 2.r2 if N ' 0 Žmod 2., and the number is Ž N q 1.r2 if N ' 1 Žmod 2.. To this we add the results of Theorems 1 and 2.
p-GROUPS
ENUMERATION OF METACYCLIC
443
3. THE CASE p s 2 The enumeration procedure for Case 2 is essentially the same as for the case p odd. We begin with the nonabelian split groups. The smallest has order 16 and belongs to Case 3 as well. Its Case 2 parameters are n s 3, t s 0, and s s 1. Its Case 3 parameters are n s 2, t s 0, and s s 1. The group has presentations ² x, y < x 8 s 1, y 2 s x 2 , yxyy1 s x 5 : , and ² x, y < x 4 s 1, y 4 s x 2 , yxyy1 s xy1 : . THEOREM 4. If N G 4 and F2 Ž N . denotes the number of nonabelian split metacyclic 2-groups of order 2 N in Case 2, then
½
F2 Ž N . s
Ž N 2 y 4 N q 4 . r4, Ž N 2 y 4 N q 3 . r4,
N ' 0 Ž mod 2 . N ' 1 Ž mod 2 . .
Proof. According to Lemma 2 we have Ny1
F2 Ž N . s
card max Ž 0, 2 n y N y 2 . , . . . , n y 3 4 .
Ý ns3
Then maxŽ0, 2 n y N y 2. s 0 if and only if n F wŽ N q 2.r2x. Therefore wŽ Nq2 .r2 x
F2 Ž N . s
Ý
Ny1
Ž n y 2. q
ns3
Ý
Ž N y n. ,
ns wŽ Nq4 .r2 x
and the result follows from Eq. Ž1.. The smallest nonsplit group in Case 2 has order 2 8 , with parameters n s 4, t s 0, s s 3, and presentation ² x, y < x 16 s 1, y 16 s x 8 , yxyy1 s x 5 : . THEOREM 5. Assume N G 8. Let G 2 Ž N . denote the number of nonsplit metacyclic 2-groups of order 2 N in Case 2, and let GŽ N . be as defined in Theorem 2. Then G2 Ž N . s
½
G Ž N . y Ž N y 4 . r2, G Ž N . y Ž N y 5 . r2,
N ' 0 Ž mod 2 . N ' 1 Ž mod 2 . .
444
STEVEN LIEDAHL
Proof. By Lemma 2 we have wŽ2 Ny3 .r3 x min Ž ny4, Nyny4 .
G2 Ž N . s
card max Ž t q 3, n y t y 2 . , . . . ,
Ý
Ý
ns4
tsmax Ž0, 2 nyNy1 .
min Ž n, N y n . y 1 4 wŽ2 Ny3 .r3 x min Ž ny3, Nyny3 .
s
card max Ž t q 2, n y t y 1 . , . . . ,
Ý
Ý
ns4
tsmax Ž1, 2 nyN .
min Ž n, N y n . y 1 4 . Then maxŽ0, 2 n y N . s 0 if and only if n F w Nr2x, therefore wŽ2 Ny3 .r3 x min Ž ny3, Nyny3 .
G2 Ž N . s
card max Ž t q 2, n y t y 1 . , . . . ,
Ý
Ý
ns3
tsmax Ž0, 2 nyN . w Nr2 x
min Ž n, N y n . y 1 4 y
Ý
min Ž n y 2, 1 .
ns3
s G Ž N . y w Nr2x q 2, and the result follows. Remark. The assumption N G 8 is used in constructing the polynomials in Theorem 5, though they correctly give the value 0 for 4 F N F 7. Similarly, the assumption N G 4 is used in constructing the polynomials in Theorem 4, though they correctly give the value 0 for 1 F N F 3. Finally we count the 2-groups in Case 3. THEOREM 6. Case 3, then
If H Ž N . denotes the number of groups of order 2 N in
HŽ N . s
½
2, Ž N 2 y 3 N q 6 . r2,
Ns3 N G 4.
Proof. By Lemma 3 we obtain each such group exactly once by letting the parameters n, t, and s vary according to n g 2, . . . , N y 1 4 , t g max Ž 0, 2 n y N y 2 . , . . . , n y 2 4 , s g n y 1, min Ž n, N y n q t q 1 . 4 ,
ENUMERATION OF METACYCLIC
p-GROUPS
445
then, if N G 5, omitting each group for which 3 F n F N y 2, t s n y 2, and s s n y 1. It follows that H Ž3. s 2 and H Ž4. s 5. For N G 5 we have Ny1
ny2
Ý
Ý
HŽ N . s
card n y 1, min Ž n, N y n q t q 1 . 4
ns2 tsmax Ž0, 2 nyNy2 .
y Ž N y 4. . Observe that cardŽ n y 1, minŽ n, N y n q t q 1.4 s 1 if and only if t s 2 n y N y 2, and maxŽ0, 2 n y N y 2. s 2 n y N y 2 if and only if n G wŽ N q 3.r2x. If N ' 0 Žmod 2. then Nr2 ny2
HŽ N . s
ns2 ts0
s
Ny1
Ý Ý 2q 1 4
Ý ns Ž Nq2 .r2
Ž N2 y 2N. q
1 4
ny2
ž
1q
Ý ts2 nyNy1
/
2 y Ž N y 4.
Ž N 2 y 4. y Ž N y 4. .
If N ' 1 Žmod 2. then Ž Nq1 .r2 ny2
HŽ N . s
Ý ns2
s
1 4
Ny1
Ý 2q ts0
Ž N 2 y 1. q
Ý ns Ž Nq3 .r2
1 4
ny2
ž
1q
/
2 y Ž N y 4.
Ý ts2 nyNy1
Ž N 2 y 2 N y 3. y Ž N y 4. .
This completes the proof. Remark. In Theorem 6 each group ² x, y < x 2 s 1, y 2 s x 2 , yxyy1 s my 1 x with n G 3, m G 2 is discounted, as the generating set Ž xy 2 , y . 2n 2m 2 ny 1 y1 y1q2 ny 1 satisfies the relations z s 1, w s z , wzw s z . This accounts for the discrepancy of N y 4 between our enumeration and that of w2, Sect. 4.2x. n
m
ny 1
y1 :
The following gives the number of metacyclic groups of order 2 N for N G 4. There are, of course, four metacyclic groups of order 8. THEOREM 7. 2 N is
If N G 4, then the number of metacyclic 2-groups of order
Ž N 3 q 48 N 2 y 168 N q 432. r72
if N ' 0 Ž mod 6 .
Ž N q 48 N y 177N q 416. r72
if N ' 1 Ž mod 6 .
Ž N 3 q 48 N 2 y 168 N q 424. r72
if N ' 2 Ž mod 6 .
Ž N 3 q 48 N 2 y 177N q 432. r72
if N ' 3 Ž mod 6 .
3
2
446
STEVEN LIEDAHL
Ž N 3 q 48 N 2 y 168 N q 416. r72
if N ' 4 Ž mod 6 .
Ž N 3 q 48 N 2 y 177N q 424. r72
if N ' 5 Ž mod 6 . .
Proof. The number of two-generator abelian 2-groups of order 2 N is Ž N q 2.r2 if N ' 0 Žmod 2., and the number is Ž N q 1.r2 if N ' 1 Žmod 2.. To this we add the results of Theorems 4, 5, and 6, then subtract 1 according to the comment which follows Lemma 3. Remark. We observe that the nonsplit groups alone account for the terms of degree 3 in Theorems 3 and 7. To conclude, we give the number of nonabelian metacyclic groups of order p N for N F 15. N ps2 p odd
3 2 1
4 5 2
5 9 4
6 15 7
7 22 10
8 32 15
9 43 20
10 57 27
11 72 34
12 91 44
13 110 53
14 134 66
15 158 78
ACKNOWLEDGMENT The author is grateful for a Research Award from the School of Natural Sciences, California State University, Fresno.
REFERENCES 1. S. Liedahl, Presentations of metacyclic p-groups with applications to K-admissibility questions, J. Algebra 169 Ž1994., 965]983. 2. W. Lindenberg, Struktur und Klassifizierung bizyklischer p-Gruppen, Ber. Gesellsch. Math. Daten¨ erarb. 40 Ž1971., 1]36. 3. M. Newman and M.-Y. Xu, Metacyclic groups of prime-power order, Ad¨ . in Math. (Beijing) 17 Ž1988., 106]107.