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2009•Journal of China West Normal UniversityRequires access

Non-normal Subgroup Are Finite Groups of q-Groups

Huaguo Shi

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Abstract

In this paper we had discussed that all non-normal subgroups are finite groups of q-groups.And we proved the following theorem:Let q be a prime,finite group G is not a Dedekind group.Then all non-normal subgroups if G are q-groups if and only if G is a non abelian q-group and G is not ismorphic to Q8×E,where Q8 is a quaternion group of order 8,E is an elementary 2-group or G=PQ,where P is a normal subgroup of G with order p,Q is a non-normal Sylow q-subgroup of G,Q is a Dedekind group and q≡1(mod p).

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What this paper is about

In this paper we had discussed that all non-normal subgroups are finite groups of q-groups.And we proved the following theorem:Let q be a prime,finite group G is not a Dedekind group.Then all non-normal subgroups if G are q-groups if and only if G is a non abelian q-group and G is not ismorphic to Q8×E,where Q8 is a quaternion group of order 8,E is an elementary 2-group or G=PQ,where P is a normal subgroup of G with order p,Q is a non-normal Sylow q-subgroup of G,Q is a Dedekind group and q≡1(mod p).

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Available abstract

In this paper we had discussed that all non-normal subgroups are finite groups of q-groups.And we proved the following theorem:Let q be a prime,finite group G is not a Dedekind group.Then all non-normal subgroups if G are q-groups if and only if G is a non abelian q-group and G is not ismorphic to Q8×E,where Q8 is a quaternion group of order 8,E is an elementary 2-group or G=PQ,where P is a normal subgroup of G with order p,Q is a non-normal Sylow q-subgroup of G,Q is a Dedekind group and q≡1(mod p).

Key concepts: Sylow theorems, Normal subgroup, Mathematics, Finite group, p-group, Index of a subgroup, Subgroup, Group (periodic table)

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