Definition. Let Γ be a group. Take A ⊂ Γ to be any subset. The subgroup generated by A, denoted 〈A〉, is the smallest subgroup of Γ containing A. Equivalently, 〈A〉 is the intersection of all subgroups of Γ containing A. Equivalently, 〈A〉 = {a1 1 · · · an n | ai ∈ A, ǫi = ±1}, the set of finite products of elements of A ∪A−1. Notation. We will frequently use Γ to denote a group. If A is a set, A ...

Finite Groups 20Dxx [1] A. Adem, J. F. Carlson, D. B. Karagueuzian, and R. James Milgram, The cohomology of the Sylow 2-subgroup of the Higman-Sims group, J. Pure Appl. Algebra 164 (2001), no. 3, 275–305. MR MR1857743 (2002g:20089) [2] Faryad Ali and Jamshid Moori, Fischer-Clifford matrices of the non-split group extension 2 · U4(2), Quaest. Math. 31 (2008), no. 1, 27–36. MR MR2404644 [3] Habib...

This survey discusses three aspects of the ways in which probability has been applied to the theory of finite groups: probabilistic statements about groups; construction of randomized algorithms in computational group theory; and application of probabilistic methods to prove deterministic theorems in group theory. It concludes with a brief summary of related results for infinite groups. Cet art...

2 Basic group theory 2 2.1 The definition of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Group homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Cosets and quotient spaces . . . . . . . . . . . . . . . . . . . . . . . . . . ....

s Prof. Paul Baginski Title: Definability and Nilpotence in Groups with Bounded Chains of Centralizers Abstract: A group G has bounded chains of centralizers (G is MC) if every chain of centralizers CG(A1) ≤ CG(A2) ≤ . . . is finite. While this class of groups is interesting in its own right, within the field of logic known as model theory, MC groups have been examined because they strictly con...

These are the notes prepared for the course MTH 751 to be offered to the PhD students at IIT Kanpur.

This paper examines two major results concerning the symmetric group, Sn. The first result, Landau’s theorem, gives an asymptotic formula for the maximum order of an element in Sn, and the second, Dixon’s Theorem, settled an open conjecture concerning the probability that two randomly selected elements of Sn will generate An or Sn.

SAGE is an open source computer algebra system implemented using an object-oriented categorical framework, with methods for objects, methods for their elements, and methods for their morphisms. Currently, SAGE has the ability to deal with abelian groups, permutation groups, and matrix groups over a finite field. This paper will present an overview of the implementations of the group-theoretical...

In Chapter 1, we introduce a notion of universality for subgroups of Polish groups that has both algebraic and topological aspects. More precisely, given a class C of subgroups of a topological group Γ, we say that a subgroup H ∈ C is a universal C subgroup of Γ if every subgroup K ∈ C is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of C with respec...