subgroup

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

• Then {{mvar|H}} is a subgroup of {{mvar|G}} if and only if {{mvar|H}} is nonempty and closed under products and inverses. Closed under products means that for every {{mvar|a}} and {{mvar|b}} in {{mvar|H}}, the product {{mvar|ab}} is in {{mvar|H}}. Closed under inverses means that for every {{mvar|a}} in {{mvar|H}}, the inverse {{math|a⁻¹}} is in {{mvar|H}}. These two conditions can be combined into one, that for every {{mvar|a}} and {{mvar|b}} in {{mvar|H}}, the element {{math|ab⁻¹}} is in {{mvar|H}}, but it is more natural and usually just as easy to test the two closure conditions separately.{{sfn|Kurzweil|Stellmacher|1998|p=4}}
• When {{mvar|H}} is finite, the test can be simplified: {{mvar|H}} is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element {{mvar|a}} of {{mvar|H}} generates a finite cyclic subgroup of {{mvar|H}}, say of order {{mvar|n}}, and then the inverse of {{mvar|a}} is {{math|aⁿ⁻¹}}.{{sfn|Kurzweil|Stellmacher|1998|p=4}}

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 {{mvar|a}} and {{mvar|b}} in {{mvar|H}}, the sum {{math|a + b}} is in {{mvar|H}}, and closed under inverses should be edited to say that for every {{mvar|a}} in {{mvar|H}}, the inverse {{math|−a}} is in {{mvar|H}}.

• The identity of a subgroup is the identity of the group: if {{mvar|G}} is a group with identity {{mvar|e_G}}, and {{mvar|H}} is a subgroup of {{mvar|G}} with identity {{mvar|e_H}}, then {{math|1=e_H = e_G}}.
• The inverse of an element in a subgroup is the inverse of the element in the group: if {{mvar|H}} is a subgroup of a group {{mvar|G}}, and {{mvar|a}} and {{mvar|b}} are elements of {{mvar|H}} such that {{math|1=ab = ba = e_H}}, then {{math|1=ab = ba = e_G}}.
• If {{mvar|H}} is a subgroup of {{mvar|G}}, then the inclusion map {{math|H → G}} sending each element {{mvar|a}} of {{mvar|H}} to itself is a homomorphism.
• The intersection of subgroups {{mvar|A}} and {{mvar|B}} of {{mvar|G}} is again a subgroup of {{mvar|G}}.{{sfn|Jacobson|2009|p=41}} For example, the intersection of the {{mvar|x}}-axis and {{mvar|y}}-axis in {{tmath|\R^2}} under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of {{mvar|G}} is a subgroup of {{mvar|G}}.
• The union of subgroups {{mvar|A}} and {{mvar|B}} is a subgroup if and only if {{math|A ⊆ B}} or {{math|B ⊆ A}}. A non-example: {{tmath|2\Z \cup 3\Z}} is not a subgroup of {{tmath|\Z,}} because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the {{mvar|x}}-axis and the {{mvar|y}}-axis in {{tmath|\R^2}} is not a subgroup of {{tmath|\R^2.}}
• If {{mvar|S}} is a subset of {{mvar|G}}, then there exists a smallest subgroup containing {{mvar|S}}, namely the intersection of all of subgroups containing {{mvar|S}}; it is denoted by {{math|{{angbr|S}}}} and is called the generating set of a group. An element of {{mvar|G}} is in {{math|{{angbr|S}}}} if and only if it is a finite product of elements of {{mvar|S}} and their inverses, possibly repeated.{{sfn|Ash|2002}}
• Every element {{mvar|a}} of a group {{mvar|G}} generates a cyclic subgroup {{math|{{angbr|a}}}}. If {{math|{{angbr|a}}}} is isomorphic to {{tmath|\Z/n\Z}} (Integers modulo n) for some positive integer {{mvar|n}}, then {{mvar|n}} is the smallest positive integer for which {{math|1=aⁿ = e}}, and {{mvar|n}} is called the order of {{mvar|a}}. If {{math|{{angbr|a}}}} is isomorphic to {{tmath|\Z,}} then {{mvar|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 {{mvar|e}} is the identity of {{mvar|G}}, then the trivial group {{math|{e} }} is the minimum subgroup of {{mvar|G}}, while the maximum subgroup is the group {{mvar|G}} itself.

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

{{Main|Coset|Lagrange's theorem (group theory)}}

Given a subgroup {{mvar|H}} and some {{mvar|a}} in {{mvar|G}}, we define the left coset {{math|1=aH = {ah : h in H}.}} Because {{mvar|a}} is invertible, the map {{math|φ : H → aH}} given by {{math|1=φ(h) = ah}} is a bijection. Furthermore, every element of {{mvar|G}} is contained in precisely one left coset of {{mvar|H}}; the left cosets are the equivalence classes corresponding to the equivalence relation {{math|a₁ ~ a₂}} if and only if {{tmath|a_1^{-1}a_2}} is in {{mvar|H}}. The number of left cosets of {{mvar|H}} is called the index of {{mvar|H}} in {{mvar|G}} and is denoted by {{math|[G : H]}}.

Lagrange's theorem states that for a finite group {{mvar|G}} and a subgroup {{mvar|H}},

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

where {{mvar|{{abs|G}}}} and {{mvar|{{abs|H}}}} denote the orders of {{mvar|G}} and {{mvar|H}}, respectively. In particular, the order of every subgroup of {{mvar|G}} (and the order of every element of {{mvar|G}}) must be a divisor of {{mvar|{{abs|G}}}}.See a [https://www.youtube.com/watch?v=TCcSZEL_3CQ didactic proof in this video].{{sfn|Dummit|Foote|2004|p=90}}

Right cosets are defined analogously: {{math|1=Ha = {ha : h in H}.}} They are also the equivalence classes for a suitable equivalence relation and their number is equal to {{math|[G : H]}}.

If {{math|1=aH = Ha}} for every {{mvar|a}} in {{mvar|G}}, then {{mvar|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 {{mvar|p}} is the lowest prime dividing the order of a finite group {{mvar|G}}, then any subgroup of index {{mvar|p}} (if such exists) is normal.

==Example: Subgroups of Z₈==

Let {{mvar|G}} be the cyclic group {{math|Z₈}} whose elements are

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

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

This group has two nontrivial subgroups: {{math|{{colorbull|orange}} J {{=}} {0, 4} }} and {{math|{{colorbull|red}} H {{=}} {0, 4, 2, 6} }}, where {{mvar|J}} is also a subgroup of {{mvar|H}}. The Cayley table for {{mvar|H}} is the top-left quadrant of the Cayley table for {{mvar|G}}; The Cayley table for {{mvar|J}} is the top-left quadrant of the Cayley table for {{mvar|H}}. The group {{mvar|G}} is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.{{sfn|Gallian|2013|p=81}}

{{math|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.

Like each group, {{math|S₄}} is a subgroup of itself.

The alternating group contains only the even permutations.

It is one of the two nontrivial proper normal subgroups of {{math|S₄}}. (The other one is its Klein subgroup.)

File:Alternating group 4; Cayley table; numbers.svg
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Each permutation {{mvar|p}} of order 2 generates a subgroup {{math|{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)

The trivial subgroup is the unique subgroup of order 1.

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

• Cartan subgroup
• Fitting subgroup
• Fixed-point subgroup
• Fully normalized subgroup
• Stable subgroup

• {{Citation| last=Jacobson| first=Nathan | author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | publisher=Dover| isbn = 978-0-486-47189-1}}.
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