• Croatian: podgrupa
• Swedish: undergrupp

• Croatian: podgrupa
• French: sous-groupe
• German: Untergruppe
• Italian: sottogruppo
• Portuguese: subgrupo
• Spanish: subgrupo
• Swedish: delgrupp

Verb

1. To divide or classify into subgroups

In group theory, given a group G under a binary operation *, we say that some subset H of G is 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 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 (i.e. H ≠ G). The trivial subgroup of any group is the subgroup 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.

In the following, we follow the usual convention of dropping * and writing the product a*b as simply ab.

Basic properties of subgroups

• H is a subgroup of the group G if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever a and b are in H, then ab and a−1 are also in H. These two conditions can be combined into one equivalent condition: whenever a and b are in H, then ab−1 is also in H.) In the case that H is finite, then H is a subgroup if and only if H is closed under products. (In this case, every element a of H generates a finite cyclic subgroup of H, and the inverse of a is then a−1 = an − 1, where n is the order of a.)
• The above condition can be stated in terms of a homomorphism; that is, H is a subgroup of a group G if and only if H is a subset of G and there is an inclusion homomorphism (i.e., i(a) = a for every a) from H to G.
• The identity of a subgroup is the identity of the group: if G is a group with identity eG, and H is a subgroup of G with identity eH, then eH = eG.
• The inverse of an element in a subgroup is the inverse of the element in the group: if H is a subgroup of a group G, and a and b are elements of H such that ab = ba = eH, then ab = ba = eG.
• The intersection of subgroups A and B is again a subgroup. The union of subgroups A and B is a subgroup if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not.
• If S is a subset of G, then there exists a minimum subgroup containing S, which can be found by taking the intersection of all of subgroups containing S; it is denoted by and is said to be the subgroup generated by S. An element of G is in if and only if it is a finite product of elements of S and their inverses.
• Every element a of a group G generates the cyclic subgroup . If is isomorphic to Z/nZ for some positive integer n, then n is the smallest positive integer for which an = e, and n is called the order of a. If is isomorphic to Z, then a is said to have infinite order.
• The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union itself.) If e is the identity of G, then the trivial group is the minimum subgroup of G, while the maximum subgroup is the group G itself.

Example

Let G be the abelian group whose elements are

G''=

and whose group operation is addition modulo eight. Its Cayley table is This group has a pair of nontrivial subgroups: J= and H=, where J is also a subgroup of H. The Cayley table for H is the top-left quadrant of the Cayley table for G. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.

Cosets and Lagrange's theorem

Given a subgroup H and some a in G, we define the left coset aH = . Because a is invertible, the map φ : H → aH given by φ(h) = ah is a bijection. Furthermore, every element of G is contained in precisely one left coset of H; the left cosets are the equivalence classes corresponding to the equivalence relation a1 ~ a2 if and only if a1−1a2 is in H. The number of left cosets of H is called the index of H in G and is denoted by [G : H].

Lagrange's theorem states that for a finite group G and a subgroup H,

[ G : H ] =

where |G| and |H| denote the orders of G and H, respectively. In particular, the order of every subgroup of G (and the order of every element of G) must be a divisor of |G|.

Right cosets are defined analogously: Ha = . They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H].

If aH = Ha for every a in G, then H is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement.

See also

• Cartan subgroup
• Fitting subgroup
• stable subgroup

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