# Subgroup

In group theory, 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 HG, 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 (i.e. HG). 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 (G,*), usually to emphasize the operation * when G carries multiple algebraic or other structures.

Read more about Subgroup:  Basic Properties of Subgroups, Cosets and Lagrange's Theorem, Example: Subgroups of Z8, Example: Subgroups of S4

### Other articles related to "subgroup, subgroups":

Special Relativity (alternative Formulations) - Very Special Relativity
... Surprisingly, Cohen and Glashow have demonstrated that a small subgroup of the Lorentz group is sufficient to explain all the current bounds ... The minimal subgroup in question can be described as follows The stabilizer of a null vector is the special Euclidean group SE(2), which contains T(2) as the subgroup of ... subgroups of the orthochronous and time-reversal respectively), is sufficient to give us all the standard predictions ...
Subgroup Test
... In abstract algebra, the one-step subgroup test is a theorem that states that for any group, a nonempty subset of that group is itself a group if the inverse of any element in the subset multiplied with any ... The two-step subgroup test is a similar theorem which requires the subset to be closed under the operation and taking of inverses ...