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.

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 {e} consisting of just the identity element.

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, it is enough that H is closed under products, since the closure under inverse follows automatically in that case.
• 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.
• 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.

Example

Let there be an abelian group (G,+) whose elements are

G={0,2,4,6,1,3,5,7}

and whose group-product is addition modulo eight. Its Cayley table is

This group has a pair of nontrivial subgroups: J={0,4} and H={0,2,4,6}, where J is also a subgroup of H. The Cayley table for H is the top-left quadrant of the Cayley table for G. 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 = {ah : h in H}. Because a is invertible, the map <math> \phi : H \rightarrow aH <math> given by <math> h \mapsto ah <math> 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

<math> [ G : H ] = { o(G) \over o(H) } <math>

where o(G) and o(H) denote the orders of G and H, respectively. In particular, if G is finite, then the order of every subgroup of G (and the order of every element of G) must be a divisor of o(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.