# Definition: Subgroup

Definition: Let $G$ be a group. $H$ is a subgroup of $G$ if $H$ is a subset of $G$ such that $H$ is also a group with the same group structure as $G$.  This means that the rule of composition, identity, and inverses in $H$ are those adopted from $G$. //

One example of this construction is as follows.  Suppose $\{0,1,2,3,4,5,6,7,8,9,10,11\}$ is the group of integers modulo 12, with addition modulo 12.  A subgroup of this group is $\{0, 6\}$, as is $\{0, 4, 8\}$, as is $\{0, 3, 6, 9\}$.  These are all subsets of the initial group that a) are closed under addition, meaning that the composition of any two elements in any of these subsets is again an element of that subset b) contain the identity element from the initial set, and c) maintains the relationship between an element and an inverse (i.e., for any element in any one of these subsets, its inverse is also in that subset).  However, the subsets $\{0, 1, 2\}$, $\{3, 6, 9\}$, and $\{2, 10, 11\}$ are not subgroups.

We introduce subgroups and discuss them in more detail in lesson 27.

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