Subgroup

Suppose that is a group, and is a subset of . For now, assume that the group operation of is written multiplicatively, denoted by juxtaposition. • Then is a subgroup of if and only if is nonempty and closed under products and inverses. Closed under products means that for every and in , the product is in . Closed under inverses means that for every in , the inverse is in . These two conditions can be combined into one, that for every and in , the element is in , but it is more natural and usually just as easy to test the two closure conditions separately. • When is finite, the test can be simplified: is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element of generates a finite cyclic subgroup of , say of order , and then the inverse of is . 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 . ==Basic properties of subgroups==

• The identity of a subgroup is the identity of the group: if is a group with identity , and is a subgroup of with identity , then . • The inverse of an element in a subgroup is the inverse of the element in the group: if is a subgroup of a group , and and are elements of such that , then . • If is a subgroup of , then the inclusion map sending each element of to itself is a homomorphism. • The intersection of subgroups and of is again a subgroup of . For example, the intersection of the -axis and -axis in under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of is a subgroup of . • The union of subgroups and is a subgroup if and only if or . A non-example: is not a subgroup of because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the -axis and the -axis in is not a subgroup of • If is a subset of , then there exists a smallest subgroup containing , namely the intersection of all of subgroups containing ; it is denoted by and is called the subgroup generated by. An element of is in if and only if it is a finite product of elements of and their inverses, possibly repeated. • Every element of a group generates a cyclic subgroup . If is isomorphic to (the integers) for some positive integer , then is the smallest positive integer for which , and is called the order of . If is isomorphic to then is said to have infinite order. • The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (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 is the identity of , then the trivial group {{math|{e} }} is the minimum subgroup of , while the maximum subgroup is the group itself. under addition. The subgroup contains only 0 and 4, and is isomorphic to \Z/2\Z. There are four left cosets of : itself, , , and (written using additive notation since this is an additive group). Together they partition the entire group into equal-size, non-overlapping sets. The index is 4. ==Cosets and Lagrange's theorem==

Given a subgroup and some in , we define the left coset {{math|1=aH = {ah : h in H}.}} 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 {{tmath|a_1^{-1}a_2}} 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 , : [ G : H ] = { |G| \over |H| } 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: {{math|1=Ha = {ha : h in H}.}} 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. ==Example: Subgroups of Z8== Let be the finite cyclic group :\mathrm{Z}_8 = \{0,1,2,3,4,5,6,7\} under addition modulo 8. The subset \{0,2,4,6\} consisting of multiples of 2 is a subgroup of \mathrm{Z}_8. More generally, for each divisor of 8, the multiples of form a subgroup. Explicitly, for d=1,2,4,8, these subgroups are \{0,1,2,3,4,5,6,7\}, \{0,2,4,6\}, \{0,4\}, \{0\}. In general, for any positive integer , one can describe all subgroups of the finite cyclic group \mathrm{Z}_n similarly: for each divisor of , the multiples of in \mathrm{Z}_n form a subgroup of order n/d, and every subgroup arises in this way. Subgroups of cyclic groups are cyclic. ==Example: Subgroups of S4==

The symmetric group is the group whose elements are the permutations of \{1,2,3,4\}. Below are all its subgroups, ordered by cardinality. 24 elements Like each group, is a subgroup of itself. 12 elements The alternating group consists of all the even permutations in . Since it is of index 2, it is a normal subgroup. 8 elements 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 1,2,3,4 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 (1234) and (24). 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 . 6 elements There are four subgroups of order 6, each isomorphic to . Each is the stabilizer of one of the elements of \{1,2,3,4\}. For example, the stabilizer of 4 is the group of permutations in that map 4 to 4, while permuting \{1,2,3\} in an arbitrary way; it is generated by the permutations (12) and (123), for instance. The four subgroups of order 6 are conjugate to each other. 4 elements There are seven subgroups of order 4, falling into three conjugacy classes of subgroups: • The subset \{1,(12)(34),(13)(24),(14)(23)\} is a normal subgroup isomorphic to the Klein four-group . • The group generated by (12) and (34) is another subgroup isomorphic to , but it is not normal. Instead it has conjugates, namely the group generated by (13) and (24) and the group generated by (14) and (23). • Each of the six 4-cycles in generates a cyclic subgroup of order 4, but each 4-cycle generates the same subgroup as its inverse, so there are only three distinct subgroups of this type. These three subgroups are conjugate to each other because all 4-cycles in are conjugate to each other. 3 elements 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. 2 elements There are nine subgroups of order 2, falling into two conjugacy classes of subgroups: • Each of the \binom{4}{2} = 6 transpositions (2-cycles) generates a subgroup of order 2. These six subgroups are conjugate. • Each of the double-transpositions (12)(34), (13)(24), (14)(23) generates a subgroup of order 2. These three subgroups are conjugate. 1 element The trivial subgroup is the unique subgroup of order 1. ==Other examples==

• The even integers form a subgroup of the integer ring the sum of two even integers is even, and the negative of an even integer is even. • Every ideal in a ring is a subgroup of the additive group of . • Every linear subspace of a vector space is a subgroup of the additive group of vectors. • In an abelian group, the elements of finite order form a subgroup called the torsion subgroup. == Notes ==