Subgroup

A non-empty subset $H$ of a group $G$ which itself is a group with respect to the operation defined on $G$. A subset $H$ of a group $G$ is a subgroup of $G$ if and only if: 1) $H$ contains the product of any two elements from $H$; and 2) $H$ contains together with any element $h$ the inverse $h^{-1}$. In the cases of finite and periodic groups, condition 2) is superfluous.

The subset of a group G consisting of the element 1 only is clearly a subgroup; it is called the unit subgroup of $G$ and is usually denoted by $E$. Also, $G$ itself is a subgroup. A subgroup different from $G$ is called a proper subgroup of $G$. A proper subgroup of an infinite group can be isomorphic to the group itself. The group $G$ itself and the subgroup $E$ are called improper subgroups of $G$, while all the others are called proper ones.

The set-theoretic intersection of any two (or any set of) subgroups of a group $G$ is a subgroup of $G$. The intersection of all subgroups of $G$ containing all elements of a certain non-empty set $M$ is called the subgroup generated by the set $M$ and is denoted by $\{M\}$. If $M$ consists of one element $a$, then $\{a\}$ is called the cyclic subgroup of the element $a$. A group that coincides with one of its cyclic subgroups is called a cyclic group.

A set-theoretic union of subgroups is, in general, not a subgroup. By the join of subgroups $H_i$, $i\in I$, one means the subgroup generated by the union of the sets $H_i$.

The product of two subsets $S_1$ and $S_2$ of a group $G$ is the set consisting of all possible (different) products $s_1{s}_2$, where $s_1\in S_1$, $s_2\in S_2$. The product of two subgroups $H_1,H_2$ is a subgroup if and only if $H_1{H}_2=H_2{H}_1$, and in that case the product $H_1{H}_2$ coincides with the subgroup generated by $H_1$ and $H_2$ (i.e. with the join of $H_1$ and $H_2$).

A homomorphic image of a subgroup is a subgroup. If a group $G_1$ is isomorphic to a subgroup $H$ of a group $G$, one says that $G_1$ can be imbedded in $G$ (as groups). If one is given two groups and each of them is isomorphic to a proper subgroup of the other, it does not necessarily follow that these groups themselves are isomorphic (cf. also Homomorphism; Isomorphism).

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