## ::Subgroup

### ::concepts

Style::group    Color::subgroup    ''H''::''g''    Table::cayley    Elements::''a''    Cyclic::thumb

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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 × H is a group operation on H. This is usually denoted HG, 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. HG). 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} ≠ HG).<ref>Hungerford (1974), p. 32</ref><ref>Artin (2011), p. 43</ref>

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.