Subgroup

Given a group G under an operation *, we say that some subset H of G is a subgroup if H is a group under the restriction of * thereto. (The same definition applies more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups.)

It is easily shown that H is a subgroup of the group G if and only if it is nonempty and closed to products and inverses. Furthermore, H's identity element is equal to G's identity element, and the inverse of an element of H is the same as the inverse of that element in G.

The subgroups of any given group form a complete lattice under inclusion. There is a minimal subgroup, the trivial group {e} (e being Gs identity element), and a maximal subgroup, the group G itself.

If S is a subset of G, then there exists a minimal subgroup containing S; it is denoted by <S> and is said to be generated by S. The elements of <S> are all finite products 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_n, +), where Z_n denotes the integers modulo n for some positive integer n (see modular arithmetic).

Order of an element of a group: Given an element x of G, the order of the cyclic subgroup <x> is called the order of x; it is the smallest positive integer n such that xⁿ = e.

Given a subgroup H and some g in G, we define the left coset g*H = {g*h : h in H}. Because g is invertible, the set g*H has just as many elements as H. 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 g₁ ~ g₂ iff g₁^-1 * g₂ 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: H*g = {h*g : h in H}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H]. If g*H = H*g for every g in G, then H is said to be a normal subgroup. In that case we can define a multiplication on cosets by

   (g1*H)*(g2*H) := (g1*g2)*H

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

The normal subgroups of any group G form a lattice under inclusion. The minimal and maximal elements are {e} and G, the greatest lower bound of two subgroup is their intersection and their least upper bound is a product group[?].