In 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 usually represented notationally by H ≤ G, read as "H is a subgroup of G".
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. 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. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.
This article will write ab for a ∗ b, as is usual.
Contents
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Basic properties of subgroups 1
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Cosets and Lagrange's theorem 2
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Example: Subgroups of Z8 3
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Example: Subgroups of S4(the symmetric group on 4 elements) 4
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12 elements 4.1
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8 elements 4.2
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6 elements 4.3
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4 elements 4.4
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3 elements 4.5
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See also 5
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Notes 6
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References 7
Basic properties of subgroups
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A subset H of the group G is a subgroup of 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 if and only if 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.)
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The above condition can be stated in terms of a homomorphism; that is, H is a subgroup of a group G if and only if H is a subset of G and there is an inclusion homomorphism (i.e., i(a) = a for every a) from H to G.
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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.
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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.
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The intersection of subgroups A and B is again a subgroup.[1] The union of subgroups A and B is a subgroup if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity.
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If S is a subset of G, then there exists a minimum subgroup containing S, which can be found by taking the intersection of all of subgroups 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.
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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.
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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
\mathbb{Z}/8\mathbb{Z}, the
integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to
\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 : H → aH 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 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 for a finite group G and a subgroup H,
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[ 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
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G=\left\{0,2,4,6,1,3,5,7\right\}
and whose group operation is
addition modulo eight. Its
Cayley table is
+
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0
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2
|
4
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6
|
1
|
3
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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 two 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.
Example: Subgroups of S4 (the symmetric group on 4 elements)
Every group has as many small subgroups as neutral elements on the main diagonal:
The
trivial group and two-element groups Z
2. These small subgroups are not counted in the following list.
12 elements
8 elements
6 elements
|
Symmetric group S 3
Subgroup:
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Symmetric group S 3
Subgroup:
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Symmetric group S 3
Subgroup:
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4 elements
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Klein four-group
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Klein four-group
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Klein four-group
|
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Cyclic group Z4
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Cyclic group Z4
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3 elements
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Cyclic group Z3
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Cyclic group Z3
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Cyclic group Z3
|
See also
Notes
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^ Jacobson (2009), p. 41
References
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