Group Theory I

“…A, then its restriction tj P (an automorphism of P) corresponds to the 1-cocycle obtained by restricting d to P=A. Now, since P is of finite p 0 -index in G, and A is a p-group, restriction-corestriction in 1-cohomology (see [10,Theorem 7.26]) shows that restriction induces an injection H 1 ðG=A; AÞ ,! H 1 ðP=A; AÞ on 1-cohomology.…”

Section: Blackburn Groups Have the Normalizer Propertymentioning

confidence: 99%

Class-preserving automorphisms and the normalizer property for Blackburn groups

Hertweck

Jespers

2009

Journal of Group Theory

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Abstract. For a group G, let U be the group of units of the integral group ring ZG. The group G is said to have the normalizer property if N U ðGÞ ¼ ZðUÞG. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being non-trivial. Groups G for which classpreserving automorphisms are inner automorphisms, Out c ðGÞ ¼ 1, have the normalizer property. Recently, Herman and Li have shown that Out c ðGÞ ¼ 1 for a finite Blackburn group G. We show that Out c ðGÞ ¼ 1 for the members G of certain classes of metabelian groups, from which the Herman-Li result follows.Together with recent work of Hertweck, Iwaki, Jespers and Juriaans, our main result implies that, for an arbitrary group G, the group Z y ðUÞ of hypercentral units of U is contained in ZðUÞG.

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“…Moreover, t(σ)1 is actually a transversal and (T, f) satisfies the conditions (a) and (b) in the proof of Theorem 4. [2,13].) The group extension G is uniquely determined by T and f .…”

Section: Theoremmentioning

confidence: 99%

On the Security of MOR Public Key Cryptosystem

Lee

Kim

Kwon³

et al. 2004

Advances in Cryptology - ASIACRYPT 2004

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Abstract. For a finite group G to be used in the MOR public key cryptosystem, it is necessary that the discrete logarithm problem(DLP) over the inner automorphism group Inn(G) of G must be computationally hard to solve. In this paper, under the assumption that the special conjugacy problem of G is easy, we show that the complexity of the MOR system over G is about log |G| times larger than that of DLP over G in a generic sense. We also introduce a group-theoretic method, called the group extension, to analyze the MOR cryptosystem. When G is considered as a group extension of H by a simple abelian group, we show that DLP over Inn(G) can be 'reduced' to DLP over Inn(H). On the other hand, we show that the reduction from DLP over Inn(G) to DLP over G is also possible for some groups. For example, when G is a nilpotent group, we obtain such a reduction by the central commutator attack.

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Group Theory I

Cited by 573 publications

References 0 publications

A note on infinite aS-groups

A note on infinite aS-groups

Class-preserving automorphisms and the normalizer property for Blackburn groups

On the Security of MOR Public Key Cryptosystem

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