BREAKING NEWS

## Summary

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called 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 × H is a group operation on H. This is often denoted HG, read as "H is a subgroup of G".

The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.

A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, HG). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).

If H is a subgroup of G, then G is sometimes called an overgroup of H.

The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups.

## Subgroup tests

Suppose that G is a group, and H is a subset of G. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition.

• Then H is a subgroup of G if and only if H is nonempty and closed under products and inverses. Closed under products means that for every a and b in H, the product ab is in H. Closed under inverses means that for every a in H, the inverse a−1 is in H. These two conditions can be combined into one, that for every a and b in H, the element ab−1 is in H, but it is more natural and usually just as easy to test the two closure conditions separately.
• When H is finite, the test can be simplified: H is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element a of H generates a finite cyclic subgroup of H, say of order n, and then the inverse of a is an−1.

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 a and b in H, the sum a + b is in H, and closed under inverses should be edited to say that for every a in H, the inverse a is in H.

## Basic properties of subgroups

• 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 H is a subgroup of G, then the inclusion map HG sending each element a of H to itself is a homomorphism.
• The intersection of subgroups A and B of G is again a subgroup of G. For example, the intersection of the x-axis and y-axis in $\mathbb {R} ^{2}$  under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.
• The union of subgroups A and B is a subgroup if and only if AB or BA. A non-example: $2\mathbb {Z} \cup 3\mathbb {Z}$  is not a subgroup of $\mathbb {Z} ,$  because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the x-axis and the y-axis in $\mathbb {R} ^{2}$  is not a subgroup of $\mathbb {R} ^{2}.$
• If S is a subset of G, then there exists a smallest subgroup containing S, namely the intersection of all of subgroups containing S; it is denoted by S and is called 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, possibly repeated.
• Every element a of a group G generates a cyclic subgroup a. If a is isomorphic to $\mathbb {Z} /n\mathbb {Z}$  (the integers mod n) 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 $\mathbb {Z} ,$  then a 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 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. G is the group Z / 8 Z , {\displaystyle \mathbb {Z} /8\mathbb {Z} ,}   the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to Z / 2 Z . {\displaystyle \mathbb {Z} /2\mathbb {Z} .}   There are four left cosets of H: H itself, 1 + H, 2 + H, and 3 + H (written using additive notation since this is an additive group). Together they partition the entire group G into equal-size, non-overlapping sets. The index [G : H] is 4.

## 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 φ : HaH given by φ(h) = ah 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 if and only if $a_{1}^{-1}a_{2}$  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 for a finite group G and a subgroup H,

$[G:H]={|G| \over |H|}$

where |G| and |H| denote the orders of G and H, respectively. In particular, the order of every subgroup of G (and the order of every element of G) must be a divisor of |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. 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 p is the lowest prime dividing the order of a finite group G, then any subgroup of index p (if such exists) is normal.

## Example: Subgroups of Z8

Let G be the cyclic group Z8 whose elements are

$G=\left\{0,4,2,6,1,5,3,7\right\}$

and whose group operation is addition modulo 8. Its Cayley table is

+ 0 4 2 6 1 5 3 7
0 0 4 2 6 1 5 3 7
4 4 0 6 2 5 1 7 3
2 2 6 4 0 3 7 5 1
6 6 2 0 4 7 3 1 5
1 1 5 3 7 2 6 4 0
5 5 1 7 3 6 2 0 4
3 3 7 5 1 4 0 6 2
7 7 3 1 5 0 4 2 6

This group has two nontrivial subgroups: J = {0, 4} and H = {0, 4, 2, 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 Cayley table for J is the top-left quadrant of the Cayley table for H. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.

## Example: Subgroups of S4

S4 is the symmetric group whose elements correspond to the permutations of 4 elements.
Below are all its subgroups, ordered by cardinality.
Each group (except those of cardinality 1 and 2) is represented by its Cayley table.

### 24 elements

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### 12 elements

The alternating group contains only the even permutations.
It is one of the two nontrivial proper normal subgroups of S4. (The other one is its Klein subgroup.)

### 2 elements

Each permutation p of order 2 generates a subgroup {1, p}. These are the permutations that have only 2-cycles:

• There are the 6 transpositions with one 2-cycle.   (green background)
• And 3 permutations with two 2-cycles.   (white background, bold numbers)

### 1 element

The trivial subgroup is the unique subgroup of order 1.

## Other examples

• The even integers form a subgroup $2\mathbb {Z}$  of the integer ring $\mathbb {Z} :$  the sum of two even integers is even, and the negative of an even integer is even.
• An ideal in a ring R is a subgroup of the additive group of R.
• A 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.