Classification of finite p-groups by their Schur multipliers

Group Theory Conference, Mashhad 2010
Ferdowsi University of Mashhad, March 10-12, 2010, p. 102-105
Classification of finite p-groups by their Schur multipliers
E. Khamseh
Islamic Azad University Shahr-e-Qods Branch, Iran
M. R. R. Moghaddam
Department of Mathematics, Ferdowsi University of Mashhad,
Mashhad, Iran
F. Saeedi
Department of Mathematics, Islamic Azad University
Mashhad–Branch, Mashhad, Iran
Abstract
Let G be a finite p-group of order p n , M(G) denotes the Schur multiplier of G. Then in
1956 Green proved that |M(G)| = p 1
2 n(n−1)−t , where t ≥ 0. Berkovich (1991), Zhou (1994)
and Ellis (1999) have determined the structure of G, when t = 0, 1, 2, 3, respectively. In 2007
Salemkar et al. classified the structure of G for t = 4 under one condition. In this paper, we
classify all abelian p-groups for t ≥ 0 and non-abelian p-groups, for 4 ≤ t ≤ 6.
Keyword and phrases: Schur multiplier, extra-special p-group
AMS subject Classification 2010:Primery 20D15; Secondary 20E34, 20F18
1 Introduction
Let G be a p-group of order p n , then J. A. Green [3] gave an upper bound p 1
2 n(n−1) for the
order of its Schur multiplier. So there exist a non-negative integer t(G) such that |M(G)| =
p 1
2 n(n−1)−t(G) . In the following theorem we collect all results which have been proved to classify
all groups, when t = 0, 1, 2, 3, 4.
Theorem 1.1 Let G be a finite p-group of order p n . Then by the above notation
(i) (Ya. G. Berkovich [1]) t(G) = 0 if and only if G is an elementary abelian p-group. Also
t(G) = 1 if and only if G ∼ = C p 2 or E 1 p 3;
(ii) (X. Zhou [8]) t(G) = 2 if and only if G ∼ = C p 2 × Cp, D8 or E 1 p 3 × Cp;
(iii) (G. Ellis [2]) t(G) = 3 if and only if G ∼ = C p 3, C p 2 × C (2)
p , D8 × C2, E 1 p 3 × C (2)
p , Q8 or E 2 p 3;
(iv) (A. R. Salemkar et al [6]) If t(G) = 4 and G is abelian, then G ∼ = C p 2 ×C p 2 or C p 2 ×C (3)
p .
Also if G is non-abelian with elementary abelian centre, then t(G) = 4 if and only if
G ∼ = D8 × C2 × C2, E 1 p 3 × C (3)
p , Q8 × C2, E 2 p 3 × Cp or X1, where
X1 = 〈a, b, c | a p2
= b p = c p = 1, [a, c] = b, [a, b] = [b, c] = 1〉
is a p-group of order p 4 with odd exponent p 2 , Cp m, D8, Q8, E 1 p 2n+1 and E 2 p 2n+1 denote the
cyclic group of order p m , dihedral and quaternion group of order 8, respectively, and the
extra-special p-groups of order p 2n+1 with odd exponent p and p 2 , respectively.
102

E. Khamseh, M. R. R. Moghaddam and F. Saeedi 103
In the present paper, using GAP programma [7] in some stages we determine all non-abelian
p-groups for which 4 ≤ t(G) ≤ 6, and by omitting the condition for the case t(G) = 4 in [6] we
have also found some new groups.
To prove the main results, we need the following results (see also [5]).
Theorem 1.2 (Schur 1907) For every finite groups H and K, we have
M(H × K) ∼ = M(H) × M(K) × H
H ′ ⊗ K
K ′
Lemma 1.1 [4] Let G be a d generator p-group of order p n , then
p 1
2 d(d−1) ≤ |G ′
||M(G)| ≤ p 1
2 n(n−1) .
Proposition 1.1 [2] Let G be a d-generator group of order p n with derived subgroup of order
p c and its central factor is a δ-generator group. Then
2 Main Results
|M(G)| ≤ p 1
2 [d(2n−2c−d−1)+2(δ−1)c]
Throughout the rest of the paper we always assume that G is a d-generator p-group of order pn with |M(G)| = p 1
2 n(n−1)−t(G) and G ′ of order pc . We also assume that G
Z(G) is a δ- generator
and the Frattini subgroup Φ(G), is of order pa . It is clear that a = n − d.
By the above notations inequality (1) implies that
Since d ≥ δ, we have
2(t − c(d + 1 − δ)) ≥ a 2 − a a ≥ c ≥ 0 (2)
(1)
2(t − c) ≥ a 2 − a. (3)
In this section, we first determine the structure of abelian p-groups G in terms of the orders
of Frattini subgroup and its Schur multiplier, for all t(G) ≥ 0. Also when G is a non-abelian
p-group we characterize the structure of G, by considering different possible choices of (c, a) in
inequality (3) when 4 ≤ t ≤ 6.
In the following result we characterize all abelian p-groups, for which t ≥ 0. Let G is a finite
abelian p-group, then we can write G ∼ = Cpn1 × ... × Cpnk × C n−(a+k)
p , where n1 + ... + nk = a + k,
if G is not elementary abelian then ni ≥ 2 and 1 ≤ k ≤ a.
Theorem 2.1 By the above notations if G be a finite abelian p-group of order p n and |Φ(G)| =
p a then
(i) t = 0 if and only if G ∼ = C (n)
p ;
(ii) t > 0 if and only if
G ∼ = Cp n 1 × ... × Cp n k × C (l)
p ,
in which l is non-negative integer equal to t
a − (a+k)2−(a+k)−2m 2a
Theorem 2.2 Let G be a non-abelian p-group of order p n , then
and |M(Cp n 1 × ... × Cp n k )| = p m .

104 Classification of finite p-groups by their . . .
(a) t(G) = 4 if and only if
G ∼ = Q8 × C2, D8 × C (2)
2 , T1, T4 (p = 2), and
G ∼ = E 1 p 3 × C (3)
p , E 2 p 3 × Cp, X1, X4, X5, X6, Y3 (p = 2);
(b) t(G) = 5 if and only if
G ∼ = Q8 × C (2)
2 , D8 × C (3)
2 , T4 × C2, T2, E 1 2 5, E 2 2 5, D16 (p = 2), and
G ∼ = E 1 p 3 × C (4)
p , E 2 p 3 × C (2)
p , E 1 p 5, E 2 p 5, X4 × Cp, X2, X7, X8, X9 (p = 2);
(c) t(G) = 6 if and only if
G ∼ = Q8 × C (3)
2 , D8 × C (4)
2 , T4 × C (2)
2 , E1 2 5 × C2, E 2 2 5 × C2, T3, (C2 × ((C4 × C2) ⋊ C2)) ⋊
C2, C2 × ((C4 × C2) ⋊ C2), QD16, Q16, C (4)
2 ⋊ C2, (C4 × C4) ⋊ C2 (p = 2), and
G ∼ = E 1 p 3 × C (5)
p , E 2 p 3 × C (3)
p , E 1 p 5 × Cp, E 2 p 5 × Cp, X4 × C (2)
p , (Cp × ((C p 2 × Cp) ⋊ Cp)) ⋊
Cp, X3, Y1, Y2, Y4 (p = 2);
where X , , ,
is, T i s and Y i s are described in tables I, II and III for groups of order p4 , for p odd,
even and groups of order p5 respectively.
TABLE I
Name Relations
X1
X2
X3
X4
X5
X6
X7
X8
X9
TABLE II
a p2 = b p = c p = 1, [a, c] = b, [a, b] = [b, c] = 1
a p2 = b p2 = 1, [a, b] = a p
a p3 = b p = 1, [a, b] = a p2
a p2 = b p = c p = 1, [b, c] = a p ,
a 9 = b 3 = c 3 = 1, [a, b] = 1, [a, c] = b, c −1 bc = a −3 b
a p = b p = c p = d p = 1, [c, d] = b, [b, d] = a, [a, b] = [a, d] = [b, c] = [a, c] = 1 (p > 3)
a p2 = b p = c p = 1, [a, c] = b, [b, c] = 1, [a, b] = a p
a p2 = b p = 1, c p = a p , [a, c] = b, [b, c] = 1, [a, b] = a p
a p2 = b p = 1, c p = a αp , [a, c] = b, [b, c] = 1, [a, b] = a p ( α = 0, 1 and non residue mod p )
Name Relations
T1 a 4 = b 2 = c 2 = 1, [a, c] = b, [a, b] = [b, c] = 1
T2
a 4 = b 4 = 1, [a, b] = a 2
T3
a 8 = b 2 = 1, [a, b] = a 4
T4 a 4 = b 2 = c 2 = 1, [b, c] = a 2 , [a, b] = [a, c] = 1
TABLE III
Name Relations
Y1 Cp × ((C
p2 × Cp) ⋊ Cp)
Y2 C
p2 × ((Cp × Cp) ⋊ Cp)
Y3
C (4)
p ⋊ Cp
Y4 (Cp × ((Cp × Cp) ⋊ Cp)) ⋊ Cp
References
[1] Berkovich, Ya. G. , On the order of the commutator subgroups and the Schur multiplier of
a finite p-group, J. Algebra 144 (1991), 269-272.
[2] Ellis, G. , On the Schur multiplier of p-groups, Comm. Algebra 27 no. 9, (1999), 4173–4177.

E. Khamseh, M. R. R. Moghaddam and F. Saeedi 105
[3] Green, J. A. , On the number of automorphism of a finite group. Pro. Royal Soc. A 237
(1956), 574–581.
[4] Jones, M. R. , Some inequalities for the multiplicator of a finite group, Pro. Amer. Math.
Soc. 39 (1973), 450–456.
[5] Karpilovsky, G. , The Schur multiplier, LMS Monogrphs New Series 2. Oxford: Oxford
University Press, 1987.
[6] Salemkar, A. R, Moghaddam, M. R. R. , Davarpanah, M. and Saeedi, F. , A remark on the
Schur multiplier of p-groups, Comm. Algebra 35 (2007), 1215–1221.
[7] The GAP Group, GAP-Groups Algorithms and Programming, Version 4.4.12, 2008
(http://www.gap-system.org/).
[8] Zhou, X. , On the order of Schur multipliers of finite p-groups, Comm. Algebra 22 no. 1,
(1994), 1–8.