The normal index of a finite group

THE NORMAL INDEX OF A FINITE GROUP. N. P. MUKHERJEE AND PRABIR BHATTACHARYA. For a maximal subgroup M of a finite group G the normal index of M is the order ...

Index of a subgroup - Wikipedia

Index of a subgroup ... measures the "relative sizes" of G and H. ... for any positive integer n. ... is infinite. ... is the set of cosets of N in G.

If H≤G is of finite index, then, we have a normal subgroup N≤G of ...

This is because in any group Z(G)⊴G and the index of Z(G) would be finite in "finite groups".. Thank you. abstract-algebra · group-theory.

THE NORMAL INDEX OF A MAXIMAL SUBGROUP OF A FINITE ...

Abstract. For a maximal subgroup M of a finite group G , the normal index r¡(G : M) is defined to be the order of a chief factor H/K where H is minimal.

On the normal index of maximal subgroups in finite groups

For a maximal subgroup M of a finite group G, the normal index of M is the order of a chief factor H/K where H is minimal in the set of ...

Existence of simultaneously normal finite index subgroups

It is well known that if K is a finite index subgroup of a group H, then there is a finite index subgroup N of K which is normal in H.

Normal subgroups of finite index in free groups - MathOverflow

Normal subgroups of finite index in free groups ... Hi all,. This is a question about the groups Hn,s introduced by Völklein in his book "Groups ...

How can the normality of a subgroup be proven when the group has ...

is a normal subgroup of finite index in G. ... I'm able to show that N is a subgroup of G by applying the subgroup test. Thing is, I'm not sure ...

On the normal indices of proper subgroups of finite groups

We extended the normal index from maximal subgroups to proper subgroups. We give a quantitative version of all results obtained by using c -normal subgroups ...

Normal subgroup of finite index - Groupprops - subwiki.org

Symbol-free definition · It is normal and its index in the whole group is finite · It is the kernel of a homomorphism to a finite group · It is the ...

On the existence of normal subgroups of prime index

Precisely, we prove that if p is a prime divisor of a finite group. G, then G has no normal subgroup of index p if and only if G = G0Gp, where Gp is the ...

(PDF) The normal index of a finite group | Nilanjan Mukherjee

Namely, if G is such a group, we call {\gamma_{0}(G)} the normal primary covering number of G; this is defined as the smallest positive integer k such that the ...

Subgroup of finite index - Groupprops

This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied. View ...

Subgroup of Finite Index Contains a Normal Subgroup of Finite Index

By the first isomorphism theorem, the quotient group G/N is isomorphic to a subgroup of Sn. In particular, G/N is a finite group, hence the index ...

Is there always a normal subgroup in a finite group G? - Quora

But a subgroup H of G of index 2 is always normal. This is because there can only be two left cosets, namely H and everything else, and there ...

the normal index of a finite group - Project Euclid

THE NORMAL INDEX OF A FINITE GROUP. N. P. MUKHERJEE AND PRABIR BHATTACHARYA. For a maximal subgroup M of a finite group G the normal index of M is the ...

THE NORMAL INDEX OF SUBGROUPS IN FINITE GROUPS

In this paper, we shall generalize the concept of the normal index and obtain the characterizations for a finite group to be solvable, p-solvable, p-nilpotent ...

Chapter 25 Finite Simple Groups

Corollary 2 [Embedding Theorem] If a finite non-Abelian simple group G has a subgroups of index n, then G is isomorphic to a subgroup of An. Proof. Let H be the ...

Normality of subgroups of prime index - Abstract Algebra

Let's begin with a well-known result about the normality of subgroups of prime index. Problem 1. Let $latex G$ be a finite group and let ...

Normal subgroups whose order and index are coprime are unique ...

Let G be a finite group, N ≤ G a normal subgroup, and suppose that | N | and [ G : N ] are relatively prime. Prove that N is the unique subgroup of order | N ...