Subgroup

In mathematics, 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.

The same definition applies 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.)
• The identity of a subgroup is the identity of the group: if G is a group with identity e_G, and H is a subgroup of G with identity e_H, then e_H = e_G.
• 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 = e_H, then ab = ba = e_G.
• If S is a subset of G, then there exists a minimum subgroup containing S; it is denoted by <S> and is said to be the subgroup generated by S. An element of G is in <S> if and only if it is a finite product of elements of S and their inverses. Groups generated by a single element are called cyclic and are isomorphic to either (Z, +), where Z denotes the integers, or to (Z/nZ, +), where Z/nZ denotes the integers modulo n for some positive integer n (see modular arithmetic).
• Every element a of a group G generates the cyclic subgroup <a>. If <a> 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 <a> is isomorphic to Z, then a is said to have infinite order.
• The subgroups of any given group form a complete lattice under inclusion. (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 {e} is the minimum subgroup of G, while the maximum subgroup is the group G itself.

Cosets and Lagrange's theorem

Given a subgroup H and some a in G, we define the left coset aH = {ah : h in H}. Because a is invertible, the map φ : H -> aH given by 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 iff 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

[G : H] |H| = |G|

where |G| and |H| denote the cardinalities of G and H, respectively. In particular, if G is finite, then the cardinality 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 = {ha : h in H}. 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. In this case, we can define a multiplication on cosets by

(a₁H)(a₂H) := (a₁a₂)H

This turns the set of cosets into a group called the quotient group G/H. There is a natural homomorphism f : G -> G/H given by f(a)=aH. The image f(H) consists only of the identity element of G/H, the coset eH = H.

In general, a group homomorphism f: G -> K sends subgroups of G to subgroups of K. Also, the preimage of any subgroup of K is a subgroup of G. We call the preimage of the trivial group {e} in K the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image f(G) of G is always isomorphic to G/ker(f). In fact, this correspondence is a bijection between the set of all quotient groups G/H and the set of all homomorphic images of G (up to isomorphism).

The normal subgroups of any group G form a lattice under inclusion. The minimum and maximum elements are {e} and G, the greatest lower bound of two subgroups is their intersection and their least upper bound is a product group.