Subgroup
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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, then H is a subgroup iff 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 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.
- 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 G be the abelian group whose elements are
- G={0,2,4,6,1,3,5,7}
and whose group operation is addition modulo eight. Its Cayley table is
| + | 0 | 2 | 4 | 6 | 1 | 3 | 5 | 7 |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 2 | 4 | 6 | 1 | 3 | 5 | 7 |
| 2 | 2 | 4 | 6 | 0 | 3 | 5 | 7 | 1 |
| 4 | 4 | 6 | 0 | 2 | 5 | 7 | 1 | 3 |
| 6 | 6 | 0 | 2 | 4 | 7 | 1 | 3 | 5 |
| 1 | 1 | 3 | 5 | 7 | 2 | 4 | 6 | 0 |
| 3 | 3 | 5 | 7 | 1 | 4 | 6 | 0 | 2 |
| 5 | 5 | 7 | 1 | 3 | 6 | 0 | 2 | 4 |
| 7 | 7 | 1 | 3 | 5 | 0 | 2 | 4 | 6 |
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. 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 = {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.de:Untergruppe fr:Sous-groupe it:Sottogruppo pl:Podgrupa fi:Aliryhmä