Subgroup

In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation on H.

Related Topics:
Group theory - Group - Binary operation - Subset - Restriction

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A proper subgroup of a group G is a subgroup H which is a proper subset of G (i.e. H ≠ G). The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G,*), usually to emphasize the operation * when G carries multiple algebraic or other structures.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In the following, we follow the usual convention of dropping * and writing the product a*b as simply ab.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~