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In this paper, using a method of construction of $1$-designs which are not necessarily symmetric, introduced by Key and Moori, we determine a number of $1$-designs with interesting parameters from the maximal subgroups and the conjugacy classes of elements of the group $PSL(2,q)$ for $q$ a power of an odd prime.

Let $ A $ be a finite-dimensional algebra with identity over the field $ \mathbb{F} $, $ U(A) $ be the group of units of $ A $ and $ L(A) $ be the set of left ideals of $ A $. It is well known that there is an equivalence relation $ \sim $ on $ L(A) $ by defining $ L_1\sim L_2\in L(A) $ if and only if there exists some $ u\in U(A) $ such that $ L_{1} = L_{2}u $. $ C(A) = \{[L]|L\in L(A)\} $ is the set of equivalence classes determined by the relation $ \sim $, which is a semigroup with respect to the natural operation $ [L_1][L_2] = [L_1L_2] $ for any $ L_1, L_2 \in L(A) $. The purpose of this paper is to describe the structures of semigroup of conjugacy classes of left ideals for the Sweedler's four-dimensional Hopf algebra $ H_{4} $.

The extended conjugate graph associated to a finite group $ G $ is defined as an undirected graph with vertex set $ G $ such that two distinct vertices joined by an edge if they are conjugate. In this article, we show that several properties of finite groups can be expressed in terms of properties of their extended conjugate graphs. In particular, we show that there is a strong connection between a graph-theoretic property, namely regularity, and an algebraic property, namely nilpotency. We then give some sufficient conditions and necessary conditions for the non-central part of an extended conjugate graph to be regular. Finally, we study extended conjugate graphs associated to groups of order $ pq $, $ p^3 $, and $ p^4 $, where $ p $ and $ q $ are distinct primes.

The aim of this work is to study the structure and sizes of conjugacy classes in certain affine semi-linear groups. This provides a wealth of finite groups of small conjugate rank that are solvable and non-nilpotent.

A subgroup H of a group G is called commensurable with a normal subgroup (cn) if there is N C G such that |HN/(H ∩ N)| is finite. We characterize generalized radical groups G which have one of the following finiteness conditions: (A) the minimal condition on non-cn subgroups of G; (B) the non-cn subgroups of G fall into finitely many conjugacy classes; (C) the non-cn subgroups of G have finite rank.

The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation.

It is proved here that if G is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then G is either locally supersoluble or a \vCernikov group. The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. As a consequence, it is shown that any infinite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble.

For a finite group $G$ let $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$. The aim of this paper is to classify all the non-nilpotent groups with $nu(G)=3$.

We prove that branch groups of automorphisms of regular rooted trees can not have polynomial conjugacy growth.

We prove that branch groups of automorphisms of regular rooted trees can not have polynomial conjugacy growth.

We present the basic results on the representation theory of the alternating groups. Our approach is based on Clifford theory.

Low density parity check (LDPC) codes are capacity-approaching codes, which means that practical constructions exist that allow the noise threshold to be set very close to the theoretical Shannon limit for a memory less channel. LDPC codes are finding increasing use in applications like LTE-Networks, digital television, high density data storage systems, deep space communication systems etc. Several algebraic and combinatorial methods are available for constructing LDPC codes. In this paper we discuss a novel low complexity algebraic method for constructing regular LDPC like codes derived from full rank codes. We demonstrate that by employing these codes over AWGN channels, coding gains in excess of 2dB over un-coded systems can be realized when soft iterative decoding using a parity check tree is employed.

12 pages, 3 figures