A Simple Classification of Finite Groups of Order p2q2 (original) (raw)

A Study on Sylow Theorems for Finding out Possible Subgroups of a Group in Different Types of Order

Mathematics and Statistics, 2022

This paper aims at treating a study on Sylow theorem of different algebraic structures as groups, order of a group, subgroups, along with the associated notions of automorphisms group of the dihedral groups, split extensions of groups and vector spaces arises from the varying properties of real and complex numbers. We must have used the Sylow theorems of this work when it's generalized. Here we discuss possible subgroups of a group in different types of order which will give us a practical knowledge to see the applications of the Sylow theorems. In algebraic structures, we deal with operations of addition and multiplication and in order structures, those of greater than, less than and so on. It is through the study of Sylow theorems that we realize the importance of some definitions as like as the exact sequences and split extensions of groups, Sylow p-subgroup and semi-direct product. Thus it has been found necessary and convenient to study these structures in detail. In situations, where it was found that a given situation satisfies the basic axioms of structure and having already known the properties of that structure. Finally, we find out possible subgroups of a group in different types of order for abelian and non-abelian cases.

A note on p-groups of order ≤ p 4

Proceedings - Mathematical Sciences, 2009

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p 4 , then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.

On capable groups of order p2q

Communications in Algebra, 2020

A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order p 2 q, for any distinct primes p, q, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, On the non abelian tensor square and capability of groups of order p 2 q, Arch. Math., 97 (2011), 299-306. We have also computed the number of distinct element centralizers of a group (finite or infinite) with central factor of order p 3 , which extends Proposition 2.

On Sylow Numbers of Some Finite Groups

we will show that if |G| = |S| and NS(G) = NS(S), where S is one of the groups: the special projective linear groups L3(q), with 5 (q −1), the projective special unitary groups U3(q), the sporadic simple groups, the alternating simple groups, and the symmetric groups of degree prime r, then G is isomorphic to S. Furthermore, we will show that if G is a finite centerless group and NS(G) = NS(L2(17)), then G is isomorphic to L2(17), and or G is isomorphic to Aut(L2(17).

On some Abelian p-groups and their capability

2015

A group is said to be capable if it is a central factor group; equivalently, if and only if a group is isomorphic to the inner automorphism group of another group. In this research, the capability of some abelian pgroups which are groups of order p4 and p5, where p is an odd prime are determined. The capability of the groups is determined by using the classifications of the groups.

Simple groups of order p · 3a · 2b

Journal of Algebra, 1970

In this paper some local group theoretic properties of a simple group G of order p * 3@ * 2b are found. These are applied in a later paper to show there are no simple groups of order 7 * 3" * 2b other than the three well-known ones. R. Brauer [4] has shown there are only the three known simple groups LI, , A, , and O,(3) of order 5 .3" * 2b. His treatment uses modular character theory especially for the prime 5. It follows from J. Thompson's X-group paper [9] that if G is a simple group of order p + 3" * 2*, then p = 5,7, 13, or 17. In our later treatment of the case p '=-T 7 we seem to need the results of the N-group paper itself. This present paper prepares the way.