subgroups

Sub Groups

Definition

A subgroup is a subset of a group that is closed under the group operation. That is, {% H %} is a subgroup if

{% H \subset G %}

and

{% h_1,h_2 \in H \rightarrow h_1h_2 \in H %}

Sub Groups Test

Test 1

Given a subset {% H %} of a group, if {% a \in H %} and {% b \in H %} implies that {% a b^{-1} \in H %}, then {% H %} is a subgroup of {% G %}.
Test 2

Given a subset {% H %} of a group, if {% a \in H %} and {% b \in H %} implies that {% a b \in H %}, and {% a*{-1} \in H %} whenever {% a \in H %}, the {% H %} is a subgroup of {% G %}.

Definitions

A subgroup {% H %} of {% G %} is called a normal subgroup if for every {% h \in H %} and every {% g \in G %}, the conjugate {% ghg^{-1} \in H %}

Additional Theorems

The center of a group is a subgroup