subgroupˈsʌbˌgrup
subgroup (n)
- plural
- subgroups
English Definitions:
subgroup (noun)
a distinct and often subordinate group within a group
subgroup (noun)
(mathematics) a subset (that is not empty) of a mathematical group
subgroup (Noun)
A group within a larger group; a group whose members are some, but not all, of the members of a larger group.
subgroup (Noun)
A subset H of a group G that is itself a group and has the same binary operation as G.
subgroup (Verb)
To divide or classify into subgroups
Subgroup
In 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 x H is a group operation on H. This is usually represented notationally by H ≤ G, read as "H is a subgroup of G". A proper subgroup of a group G is a subgroup H which is a proper subset of G. The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. 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, usually to emphasize the operation * when G carries multiple algebraic or other structures. This article will write ab for a*b, as is usual.
Subgroup
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 often 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 (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).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.
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