In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group under a binary operation âÂÂ, a subset of is called a subgroup of if also forms a group under the operation âÂÂ. More precisely, is a subgroup of if the restriction of â to is a group operation on . This is often denoted , read as " is a subgroup of ".
The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.
A proper subgroup of a group is a subgroup which is a proper subset of (that is, ). This is often represented notationally by , read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is, ).
If is a subgroup of , then is sometimes called an overgroup of .
The same definitions apply more generally when is an arbitrary semigroup, but this article will only deal with subgroups of groups.
Suppose that is a group, and is a subset of . For now, assume that the group operation of is written multiplicatively, denoted by juxtaposition.
If the group operation is instead denoted by addition, then closed under products should be replaced by closed under addition, which is the condition that for every and in , the sum is in , and closed under inverses should be edited to say that for every in , the inverse is in .
Given a subgroup and some in , we define the left coset Because is invertible, the map given by is a bijection. Furthermore, every element of is contained in precisely one left coset of ; the left cosets are the equivalence classes corresponding to the equivalence relation if and only if is in . The number of left cosets of is called the index of in and is denoted by .
Lagrange's theorem states that for a finite group and a subgroup ,
where and denote the orders of and , respectively. In particular, the order of every subgroup of (and the order of every element of ) must be a divisor of .
Right cosets are defined analogously: They are also the equivalence classes for a suitable equivalence relation and their number is equal to .
If for every in , then 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. More generally, if is the lowest prime dividing the order of a finite group , then any subgroup of index (if such exists) is normal.
Let be the finite cyclic group
under addition modulo 8. The subset consisting of multiples of 2 is a subgroup of . More generally, for each divisor of 8, the multiples of form a subgroup. Explicitly, for , these subgroups are .
In general, for any positive integer , one can describe all subgroups of the finite cyclic group similarly: for each divisor of , the multiples of in form a subgroup of order , and every subgroup arises in this way.
Subgroups of cyclic groups are cyclic.
The symmetric group is the group whose elements are the permutations of .<br> Below are all its subgroups, ordered by cardinality.<br>
Like each group, is a subgroup of itself.
The alternating group consists of all the in . Since it is of index 2, it is a normal subgroup.
There are three subgroups of order 8, each isomorphic to the dihedral group , the group of symmetries of a square.
Labeling the vertices of a square clockwise lets one view as a subgroup of . This subgroup is generated by the 90-degree clockwise rotation and by the reflection in the diagonal axis joining vertices 1 and 3; these are the permutations and .
Up to symmetries of the square, there are three different ways to label the vertices of a square, distinguished by which pairs of numbers appear on opposite corners. In the labeling above, 1 and 3 were opposite, and 2 and 4 were opposite; another choice has 1 and 4 opposite, and 2 and 3 opposite; the third choice has 1 and 2 opposite, and 3 and 4 opposite. The three labelings give rise to three different subgroups of order 8 in , conjugate to each other, each isomorphic to .
There are four subgroups of order 6, each isomorphic to . Each is the stabilizer of one of the elements of . For example, the stabilizer of 4 is the group of permutations in that map 4 to 4, while permuting in an arbitrary way; it is generated by the permutations and , for instance. The four subgroups of order 6 are conjugate to each other.
There are seven subgroups of order 4, falling into three conjugacy classes of subgroups:
There are four subgroups of order 3, each generated by a 3-cycle. There are eight 3-cycles in , but each generates the same subgroup as its inverse. The resulting four subgroups are conjugate to each other.
There are nine subgroups of order 2, falling into two conjugacy classes of subgroups:
The trivial subgroup is the unique subgroup of order 1.