Subgroup
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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 Template:Math is a group operation on Template:Mvar. This is often denoted Template:Math, 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.Template:Sfn
A proper subgroup of a group Template:Mvar is a subgroup Template:Mvar which is a proper subset of Template:Mvar (that is, Template:Math). This is often represented notationally by Template:Math, read as "Template:Mvar is a proper subgroup of Template:Mvar". Some authors also exclude the trivial group from being proper (that is, Template:Math).Template:SfnTemplate:Sfn
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
[edit]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 Template:Math 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 Template:Math is in Template:Mvar, but it is more natural and usually just as easy to test the two closure conditions separately.Template:Sfn
- 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 Template:Math.Template:Sfn
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 Template:Math is in Template:Mvar, and closed under inverses should be edited to say that for every Template:Mvar in Template:Mvar, the inverse Template:Math is in Template:Mvar.
Basic properties of subgroups
[edit]- 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 Template:Math.
- 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 Template:Math, then Template:Math.
- If Template:Mvar is a subgroup of Template:Mvar, then the inclusion map Template:Math 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.Template:Sfn 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 Template:Math or Template:Math. 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:Math and is called the [[generating set of a group|subgroup generated by Template:Mvar]]. An element of Template:Mvar is in Template:Math if and only if it is a finite product of elements of Template:Mvar and their inverses, possibly repeated.Template:Sfn
- Every element Template:Mvar of a group Template:Mvar generates a cyclic subgroup Template:Math. If Template:Math is isomorphic to Template:Tmath ([[Integers modulo n|the integers Template:Math]]) for some positive integer Template:Mvar, then Template:Mvar is the smallest positive integer for which Template:Math, and Template:Mvar is called the order of Template:Mvar. If Template:Math 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 Template:Math is the minimum subgroup of Template:Mvar, while the maximum subgroup is the group Template:Mvar itself.
Cosets and Lagrange's theorem
[edit]Template:Main Given a subgroup Template:Mvar and some Template:Mvar in Template:Mvar, we define the left coset Template:Math Because Template:Mvar is invertible, the map Template:Math given by Template:Math 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 Template:Math 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 Template:Math.
Lagrange's theorem states that for a finite group Template:Mvar and a subgroup Template:Mvar,
- <math> [ G : H ] = { |G| \over |H| }</math>
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.<ref>See a didactic proof in this video.</ref>Template:Sfn
Right cosets are defined analogously: Template:Math They are also the equivalence classes for a suitable equivalence relation and their number is equal to Template:Math.
If Template:Math 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
[edit]Let Template:Mvar be the cyclic group Template:Math whose elements are
- <math>G = \left\{0, 4, 2, 6, 1, 5, 3, 7\right\}</math>
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: Template:Math and Template:Math, where Template:Mvar is also a subgroup of Template:Mvar. The Cayley table for Template:Mvar is the top-left quadrant of the Cayley table for Template:Mvar; The Cayley table for Template:Mvar is the top-left quadrant of the Cayley table for Template:Mvar. The group Template:Mvar is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.Template:Sfn
Example: Subgroups of S4Template:Anchor
[edit]Template:Math 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
[edit]Like each group, Template:Math is a subgroup of itself.
12 elements
[edit]The alternating group contains only the even permutations.
It is one of the two nontrivial proper normal subgroups of Template:Math. (The other one is its Klein subgroup.)
Subgroups:
File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg
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File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svgFile:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg
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8 elements
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6 elements
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4 elements
[edit]3 elements
[edit]2 elements
[edit]Each permutation Template:Mvar of order 2 generates a subgroup Template:Math}.
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
[edit]The trivial subgroup is the unique subgroup of order 1.
Other examples
[edit]- 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.
- An ideal in a ring Template:Mvar is a subgroup of the additive group of Template:Mvar.
- 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.
See also
[edit]Notes
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