Subgroup
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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 usually denoted H ≤ G, 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 (i. e. H ≠ G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (i. e. {e} ≠ H ≠ G).[1][2]
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.
Basic properties of subgroups
- 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.)
- 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.
- 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.
- The intersection of subgroups A and B is again a subgroup.[3] 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.
- 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.
- 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.
- 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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Left_cosets_of_Z_2_in_Z_8.svg/220px-Left_cosets_of_Z_2_in_Z_8.svg.png)
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,
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|.[4][5]
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
and whose group operation is addition modulo eight. Its Cayley table is
+ | 0 | 2 | 4 | 6 | 1 | 3 | 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 Z2. These small subgroups are not counted in the following list.
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Hasse diagrams of the lattice of subgroups of S4 |
12 elements
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Alternating_group_4%3B_Cayley_table%3B_numbers.svg/323px-Alternating_group_4%3B_Cayley_table%3B_numbers.svg.png)
Subgroups:
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/70px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png)
8 elements
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6 elements
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4 elements
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3 elements
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Other examples
- The even integers are a subgroup of the additive group of integers: when you add two even numbers, you get an even number.
- An ideal in a ring is a subgroup of the additive group of .
- A linear subspace of a vector space is a subgroup of the additive group of vectors.
- Let be an abelian group; the elements of that have finite period form a subgroup of called the torsion subgroup of .
See also
Notes
- ^ Hungerford (1974), p. 32
- ^ Artin (2011), p. 43
- ^ Jacobson (2009), p. 41
- ^ See a didactic proof in this video.
- ^ S., Dummit, David (2004). Abstract algebra. Foote, Richard M., 1950- (3. ed.). Hoboken, NJ: Wiley. p. 90. ISBN 9780471452348. OCLC 248917264.
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References
- Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1.
- Hungerford, Thomas (1974), Algebra (1st ed.), Springer-Verlag, ISBN 9780387905181.
- Artin, Michael (2011), Algebra (2nd ed.), Prentice Hall, ISBN 9780132413770.
- S., Dummit, David (2004). Abstract algebra. Foote, Richard M., 1950- (3. ed.). Hoboken, NJ: Wiley. ISBN 9780471452348. OCLC 248917264.
{{cite book}}
: CS1 maint: multiple names: authors list (link)