## ::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 *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*).<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.

This article will write *ab* for *a* ∗ *b*, as is usual.

**Subgroup sections**

Intro Basic properties of subgroups Cosets and Lagrange's theorem See also Notes References

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