Subgroup

SG#DEF. Subgroup.

Let $(G,*)$ be a group. $(H,*)$ is a subgroup of $G$ if it is a group and $H\subseteq G$.

SG#PROP-NEU.

Let $G$ be a group with neutral element $e$ and $H$ be a subgroup. Then the neutral element of $H$ is also $e$.

Proof.If $e’$ is the neutral element of $H$,

Since a group is cancellative (GRP#PROP-CAN),

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SG#PROP-A.

Let $G$ be a group and $H\subseteq G$ be non-empty. $H$ is a subgroup as soon as:

1. Closure under multiplication. For all $a$, $b\in H$,

   \[ab \in H.\]

2. Closure under inversion. For all $a\in H$,

   \[a^{-1} \in H.\]

SG#PROP-B.

Let $G$ be a group and $H\subseteq G$ be non-empty. $H$ is a subgroup as soon as for all $a$, $b\in H$,

\[ab^{-1} \in H.\]

SG#PROP-C.

Let $G$ be a finite group and $H\subseteq G$ be non-empty. $H$ is a subgroup as soon as $H$ is closed under multiplication, i.e for all $a$, $b\in H$,

\[ab \in H.\]