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Symmetric generation of M₂₂
This study will prove the Mathieu group M₂₂ contains two symmetric generating sets with control grougp L₃ (2). The first generating set consists of order 3 elements while the second consists of involutions
Coprime invariable generation and minimal-exponent groups
A finite group is \emph{coprimely-invariably generated} if there exists a
set of generators of with the property that the orders
are pairwise coprime and that for all
the set generates .
We show that if is coprimely-invariably generated, then can be
generated with three elements, or two if is soluble, and that has zero
presentation rank. As a corollary, we show that if is any finite group such
that no proper subgroup has the same exponent as , then has zero
presentation rank. Furthermore, we show that every finite simple group is
coprimely-invariably generated.
Along the way, we show that for each finite simple group , and for each
partition of the primes dividing , the product of the
number of conjugacy classes of -elements satisfies
$\prod_{i=1}^u k_{\pi_i}(S) \leq \frac{|S|}{2| Out S|}.
Constructions in public-key cryptography over matrix groups
ISBN : 978-0-8218-4037-5International audienceThe purpose of the paper is to give new key agreement protocols (a multi-party extension of the protocol due to Anshel-Anshel-Goldfeld and a generalization of the Diffie-Hellman protocol from abelian to solvable groups) and a new homomorphic public-key cryptosystem. They rely on difficulty of the conjugacy and membership problems for subgroups of a given group. To support these and other known cryptographic schemes we present a general technique to produce a family of instances being matrix groups (over finite commutative rings) which play a role for these schemes similar to the groups in the existing cryptographic constructions like RSA or discrete logarithm
Infinite products of finite simple groups
We classify those sequences of
finite simple nonabelian groups such that the full product
has property (FA).Comment: AMS-LaTex file, 44 pages. To appear in Tran. Amer. Math. So
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