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subgroups

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Definitions of subgroups in various dictionaries:

noun - a distinct and often subordinate group within a group

noun - (mathematics) a subset (that is not empty) of a mathematical group

verb - to divide into smaller groups

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Subgroups description
In group theory, a branch of mathematics, given a group G under a binary operation , a subset H of G is called 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 × H is a group operation on H. This is usually denoted H G, read as "H is a subgroup of G". * The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. * A proper subgroup of a group G is a subgroup H which is a proper subset of G (i. e. H G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (i. e. {e} H G).If H is a subgroup of G, then G is sometimes called an overgroup of H. * 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 carri

Subgroups description

In group theory, a branch of mathematics, given a group G under a binary operation , a subset H of G is called 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 × H is a group operation on H. This is usually denoted H G, read as "H is a subgroup of G".
* The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.
* A proper subgroup of a group G is a subgroup H which is a proper subset of G (i. e. H G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (i. e. {e} H G).If H is a subgroup of G, then G is sometimes called an overgroup of H.
* 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 carri

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