Subgroup

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 Template:Mvar under a binary operation ∗, a subset Template:Mvar of Template:Mvar is called a subgroup of Template:Mvar if Template:Mvar also forms a group under the operation ∗. More precisely, Template:Mvar is a subgroup of Template:Mvar if the restriction of ∗ to H × H is a group operation on Template:Mvar. This is often denoted H ≤ G, read as "Template:Mvar is a subgroup of Template:Mvar".

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

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

If Template:Mvar is a subgroup of Template:Mvar, then Template:Mvar is sometimes called an overgroup of Template:Mvar.

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

Subgroup tests

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

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

Basic properties of subgroups

• The identity of a subgroup is the identity of the group: if Template:Mvar is a group with identity Template:Mvar, and Template:Mvar is a subgroup of Template:Mvar with identity Template:Mvar, then e_H = e_G.
• The inverse of an element in a subgroup is the inverse of the element in the group: if Template:Mvar is a subgroup of a group Template:Mvar, and Template:Mvar and Template:Mvar are elements of Template:Mvar such that ab = ba = e_H, then ab = ba = e_G.
• If Template:Mvar is a subgroup of Template:Mvar, then the inclusion map H → G sending each element Template:Mvar of Template:Mvar to itself is a homomorphism.
• The intersection of subgroups Template:Mvar and Template:Mvar of Template:Mvar is again a subgroup of Template:Mvar. For example, the intersection of the Template:Mvar-axis and Template:Mvar-axis in Template:Tmath under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of Template:Mvar is a subgroup of Template:Mvar.
• The union of subgroups Template:Mvar and Template:Mvar is a subgroup if and only if A ⊆ B or B ⊆ A. A non-example: Template:Tmath is not a subgroup of Template:Tmath because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the Template:Mvar-axis and the Template:Mvar-axis in Template:Tmath is not a subgroup of Template:Tmath
• If Template:Mvar is a subset of Template:Mvar, then there exists a smallest subgroup containing Template:Mvar, namely the intersection of all of subgroups containing Template:Mvar; it is denoted by Template:Angbr and is called the [[generating set of a group|subgroup generated by Template:Mvar]]. An element of Template:Mvar is in Template:Angbr if and only if it is a finite product of elements of Template:Mvar and their inverses, possibly repeated.
• Every element Template:Mvar of a group Template:Mvar generates a cyclic subgroup Template:Angbr. If Template:Angbr is isomorphic to Template:Tmath (the integers mod n) for some positive integer Template:Mvar, then Template:Mvar is the smallest positive integer for which aⁿ = e, and Template:Mvar is called the order of Template:Mvar. If Template:Angbr is isomorphic to Template:Tmath then Template:Mvar 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 Template:Mvar is the identity of Template:Mvar, then the trivial group {e} is the minimum subgroup of Template:Mvar, while the maximum subgroup is the group Template:Mvar itself.

File:Left cosets of Z 2 in Z 8.svg

Template:Mvar is the group $\mathbb{Z}/8\mathbb{Z},$ the integers mod 8 under addition. The subgroup Template:Mvar contains only 0 and 4, and is isomorphic to $\mathbb{Z}/2\mathbb{Z}.$ There are four left cosets of Template:Mvar: Template:Mvar itself, 1 + H, 2 + H, and 3 + H (written using additive notation since this is an additive group). Together they partition the entire group Template:Mvar into equal-size, non-overlapping sets. The index [G : H] is 4.

Cosets and Lagrange's theorem

Given a subgroup Template:Mvar and some Template:Mvar in Template:Mvar, we define the left coset aH = {ah : h in H}. Because Template:Mvar is invertible, the map φ : H → aH given by φ(h) = ah is a bijection. Furthermore, every element of Template:Mvar is contained in precisely one left coset of Template:Mvar; the left cosets are the equivalence classes corresponding to the equivalence relation a₁ ~ a₂ if and only if Template:Tmath is in Template:Mvar. The number of left cosets of Template:Mvar is called the index of Template:Mvar in Template:Mvar and is denoted by [G : H].

Lagrange's theorem states that for a finite group Template:Mvar and a subgroup Template:Mvar,

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

where Template:Mvar and Template:Mvar denote the orders of Template:Mvar and Template:Mvar, respectively. In particular, the order of every subgroup of Template:Mvar (and the order of every element of Template:Mvar) must be a divisor of Template:Mvar.

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 Template:Mvar in Template:Mvar, then Template:Mvar 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 Template:Mvar is the lowest prime dividing the order of a finite group Template:Mvar, then any subgroup of index Template:Mvar (if such exists) is normal.

Example: Subgroups of Z₈

Let Template:Mvar 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 Template:Mvar of 8, the multiples of Template:Mvar 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 Template:Mvar, one can describe all subgroups of the finite cyclic group ${\mathrm{Z}}_n$ similarly: for each divisor Template:Mvar of Template:Mvar, the multiples of Template:Mvar 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 S₄ 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, S₄ is a subgroup of itself.

12 elements

The alternating group A₄ consists of all the even permutations in S₄. 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 D₄, the group of symmetries of a square.

Labeling the vertices of a square $1,2,3,4$ clockwise lets one view D₄ as a subgroup of S₄. 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 S₄, conjugate to each other, each isomorphic to D₄.

6 elements

There are four subgroups of order 6, each isomorphic to S₃. 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 S₄ 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 V₄.

• The group generated by $(12)$ and $(34)$ is another subgroup isomorphic to V₄, 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 S₄ 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 S₄ 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 S₄, 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 $(\genfrac{}{}{0ex}{}{4}{2})=6$ 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 Template:Tmath of the integer ring Template:Tmath the sum of two even integers is even, and the negative of an even integer is even.
• Every ideal in a ring Template:Mvar is a subgroup of the additive group of Template:Mvar.
• 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.

References

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