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Almost all one-relator groups with at least three generators are residually finite

Mark V. Sapir, Iva Špakulová (2011)

Journal of the European Mathematical Society

We prove that with probability tending to 1, a one-relator group with at least three generators and the relator of length n is residually finite, is a virtually residually (finite p -)group for all sufficiently large p , and is coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions.

Almost fixed-point-free automorphisms of prime order

Bertram Wehrfritz (2011)

Open Mathematics

Let ϕ be an automorphism of prime order p of the group G with C G(ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are...

Basic subgroups in abelian group rings

Peter Vassilev Danchev (2002)

Czechoslovak Mathematical Journal

Suppose R is a commutative ring with identity of prime characteristic p and G is an arbitrary abelian p -group. In the present paper, a basic subgroup and a lower basic subgroup of the p -component U p ( R G ) and of the factor-group U p ( R G ) / G of the unit group U ( R G ) in the modular group algebra R G are established, in the case when R is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed p -component S ( R G ) and of the quotient group S ( R G ) / G p are given when R is perfect and G is arbitrary whose G / G p is p -divisible....

Basic subgroups in commutative modular group rings

Peter Vassilev Danchev (2004)

Mathematica Bohemica

Let S ( R G ) be a normed Sylow p -subgroup in a group ring R G of an abelian group G with p -component G p and a p -basic subgroup B over a commutative unitary ring R with prime characteristic p . The first central result is that 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) is basic in S ( R G ) and B [ 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) ] is p -basic in V ( R G ) , and [ 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) ] G p / G p is basic in S ( R G ) / G p and [ 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) ] G / G is p -basic in V ( R G ) / G , provided in both cases G / G p is p -divisible and R is such that its maximal perfect subring R p i has no nilpotents whenever i is natural. The second major result is that B ( 1 + I ( R G ; B p ) ) is p -basic in V ( R G ) and ( 1 + I ( R G ; B p ) ) G / G is p -basic in V ( R G ) / G ,...

Bounding the orders of finite subgroups.

Ian J. Leary, Brita E. A. Nucinkis (2001)

Publicacions Matemàtiques

We give homological conditions on groups such that whenever the conditions hold for a group G, there is a bound on the orders of finite subgroups of G. This extends a result of P. H. Kropholler. We also suggest a weaker condition under which the same conclusion might hold.

Cayley orders

Arjeh M. Cohen, Gabriele Nebe, Wilhelm Plesken (1996)

Compositio Mathematica

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