Powerful p-groups have non-inner automorphisms of order p and some cohomology

Abdollahi, Aliréza

doi:10.1016/j.jalgebra.2009.10.013

Cited by 32 publications

(39 citation statements)

References 9 publications

(21 reference statements)

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“…So in particular if G is a nonabelian regular p-group and is the Frattini subgroup of G, then H n (G/ , Z( )) = 0 for all n; Schmid then asks whether this result holds more generally. Abdollahi [1] has given some cases where the result holds, and in the final section we prove the following result.…”

Section: Theorem 2 Let G Be a Finite P K -Central P-group Of Coclassmentioning

confidence: 55%

“…By the result of Uchida [24] we know that A is cohomologically trivial if H r (Q, A) = 0 for just one integer r. In [20] Schmid investigates when the cohomology is non-trivial, he proves that if G is a regular p-group and Q = G/N is not cyclic then H n (Q, Z(N)) = 0 for all n. So, in particular, if G is a non-abelian regular p-group and is the Frattini subgroup of G then H n (G/ , Z( )) = 0 for all n, Schmid then asks whether this holds more generally. Abdollahi addresses this question in [1] (and uses the alternative definition of p-central as mentioned in our Introduction) and poses the following more general question: By Uchida's result we will be able to restrict our attention to H 0 (Q, Z(N)). Recall, H 0 (Q, A) = A Q /A τ , where A Q denotes the fixed points of A under the action of Q, and A τ denotes the image of A under the trace map τ = τ Q .…”

Section: Theorem 1 Let G Be a Finite Group Of Exponent P And Nilpotementioning

confidence: 99%

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A NOTE ONp-CENTRAL GROUPS

Camina¹,

Thillaisundaram²

2013

Glasgow Math. J.

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A group G is n-central if Gn ≤ Z(G), that is the subgroup of G generated by n-powers of G lies in the centre of G. We investigate pk-central groups for p a prime number. For G a finite group of exponent pk, the covering group of G is pk-central. Using this we show that the exponent of the Schur multiplier of G is bounded by $p^{\lceil \frac{c}{p-1} \rceil}$, where c is the nilpotency class of G. Next we give an explicit bound for the order of a finite pk-central p-group of coclass r. Lastly, we establish that for G, a finite p-central p-group, and N, a proper non-maximal normal subgroup of G, the Tate cohomology Hn(G/N, Z(N)) is non-trivial for all n. This final statement answers a question of Schmid concerning groups with non-trivial Tate cohomology.

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Section: Theorem 2 Let G Be a Finite P K -Central P-group Of Coclassmentioning

confidence: 55%

Section: Theorem 1 Let G Be a Finite Group Of Exponent P And Nilpotementioning

confidence: 99%

A NOTE ONp-CENTRAL GROUPS

Camina¹,

Thillaisundaram²

2013

Glasgow Math. J.

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“…Indeed, Deaconescu [12] proved it for all finite p-groups G which are not strongly Frattinian. Moreover, Abdollahi [2] proved it for finite p-groups G such that G/Z(G) is a powerful p-group, and Jamali and Visesh [14] did the same for finite p-groups with cyclic commutator subgroup. In the realm of finite groups, quite recently, the result has been confirmed for semi-abelian p-groups by Benmoussa and Guerboussa [6], and for p-groups of nilpotency class 3, by Abdollahi, Ghoraishi and Wilkens [3].…”

Section: Introductionmentioning

confidence: 94%

“…Furthermore, as a result due to Abdollahi in [2], we know that if G is a finite p-group such that G has no non-inner automorphisms of order p leaving Φ(G) elementwise fixed, then d(Z 2 (G)/Z(G)) = d(G)d(Z(G)). Thus, in view of this previous matter, we may assume that the condition d(Z 2 (G)/Z(G)) = d(G)d(Z(G)) holds.…”

Section: Berkovich Conjecture For Two Generator Normally Constrained mentioning

confidence: 99%

A note on non-inner automorphisms in finite normally constrained $p$-groups

Gavioli¹,

Legarreta²,

Ruscitti³

2019

Publ. Math. Debrecen

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“…The conjecture has been established for p-groups of class 2 and 3 [2,3,11], for regular p-groups [13], for p-groups G in which G/Z(G) is powerful [1], for p-groups G in which (G, Z(G)) is a Camina pair and p 2 [9], for 2-groups with a cyclic commutator subgroup [10], and for p-groups of order p m and exponent p m−2 [14]. It is worth noting that most of the noninner automorphisms given in these results leave either Φ(G) or Z(G) elementwise fixed.…”

Section: Introductionmentioning

confidence: 99%