Definition – Subgroup – Math-reference.net

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2. Theorems

Definition – Subgroup

Let $G$ be a group, and let $R\subseteq G$ . If $R$ is also a group (under the group axioms), then we say that $R$ is a subgroup of $G$ .

Types of Subgroup

Definition 1 – If $G$ is any group with a $R$ a subgroup of $G$ , and if $R$ is a cyclic group, then we say that $R$ is a (cyclic) subgroup of $G$ . Of course, we may still refer to $R$ simply as a subgroup of $G$ .

Definition 2 – If $H$ is a subgroup of $G$ , then we refer to $H$ as a normal subgroup of $G$ if

$\forall (x\in H\text{,} ~y\in G)\text{,} ~yxy^{-1} \in H$

Ie, if $H$ is closed under conjugates, where.