4.Subgroup

definition

Let $<G,\ast >$ be a group, if:

(1) $S$ is a nonempty subset of $G$;

(2) $<S,\ast >$ is a group.

Then $<S,\ast >$ is a subgroup of $<G,\ast >$, denoted as $S\le G$​.

All group $G$ have subgroup $\{e\}$ and $G$, called Trivial subgroup, the nontrivial subgroup of G is called the proper subgroup, denoted as $S<G$​.

Determination of subgroups

Let $S$ be a nonempty subset of group $G$, then the following three propositions are equivalent:

(1) $S$ is a subgroup of $G$.

(2) For $\mathrm{\forall }a,b\in S$, have $ab\in S$ and $a^{-1}\in S$ (The first subgroup criterion theorem).

(3) For $\mathrm{\forall }a,b\in S$, have $a{b}^{-1}\in S$​ (The second subgroup criterion theorem).

Determination of finite subgroups

(The finite subgroup decision theorem)

Let $S$ be a finite nonempty subset of group $G$, then the necessary and sufficient condition for $S$ to be a subgroup of $G$ is:

$$\mathrm{\forall }a,b\in S,ab\in S$$

Conservatism of subgroups

Let $G$ be a group, $H_1,H_2$ are two subgroup of $G$(which means $H_1\le G,H_2\le G$), then:

(1) $H_1\cap H_2\le G$.

(2) $H_1\cup H_2\le G\iff H_1\subseteq H_2$ or $H_2\subseteq H_1$​.

The subgroups of $G$ must have inclusion relations, so that their union can be a subgroup of $G$. There must be an inclusion relationship between subgroups of $G$.