Ingroup theory, a branch ofmathematics, 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 agroupG under abinary operation ∗, asubsetH ofG is called asubgroup ofG ifH also forms a group under the operation ∗. More precisely,H is a subgroup ofG if therestriction of ∗ toH ×H is a group operation onH. This is often denotedH ≤G, read as "H is a subgroup ofG".
Thetrivial subgroup of any group is the subgroup {e} consisting of just the identity element.[1]
Aproper subgroup of a groupG is a subgroupH which is aproper subset ofG (that is,H ≠G). This is often represented notationally byH <G, read as "H is a proper subgroup ofG". Some authors also exclude the trivial group from being proper (that is,H ≠ {e}).[2][3]
IfH is a subgroup ofG, thenG is sometimes called anovergroup ofH.
The same definitions apply more generally whenG is an arbitrarysemigroup, but this article will only deal with subgroups of groups.
Suppose thatG is a group, andH is a subset ofG. For now, assume that the group operation ofG is written multiplicatively, denoted by juxtaposition.
ThenH is a subgroup ofGif and only ifH is nonempty andclosed under products and inverses.Closed under products means that for everya andb inH, the productab is inH.Closed under inverses means that for everya inH, the inversea−1 is inH. These two conditions can be combined into one, that for everya andb inH, the elementab−1 is inH, but it is more natural and usually just as easy to test the two closure conditions separately.[4]
WhenH isfinite, the test can be simplified:H is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every elementa ofH generates a finite cyclic subgroup ofH, say of ordern, and then the inverse ofa isan−1.[4]
If the group operation is instead denoted by addition, thenclosed under products should be replaced byclosed under addition, which is the condition that for everya andb inH, the suma +b is inH, andclosed under inverses should be edited to say that for everya inH, the inverse−a is inH.
Theidentity of a subgroup is the identity of the group: ifG is a group with identityeG, andH is a subgroup ofG with identityeH, theneH =eG.
Theinverse of an element in a subgroup is the inverse of the element in the group: ifH is a subgroup of a groupG, anda andb are elements ofH such thatab =ba =eH, thenab =ba =eG.
IfH is a subgroup ofG, then the inclusion mapH →G sending each elementa ofH to itself is ahomomorphism.
Theintersection of subgroupsA andB ofG is again a subgroup ofG.[5] For example, the intersection of thex-axis andy-axis in under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups ofG is a subgroup ofG.
Theunion of subgroupsA andB is a subgroup if and only ifA ⊆B orB ⊆A. A non-example: is not a subgroup of because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of thex-axis and they-axis in is not a subgroup of
IfS is a subset ofG, then there exists a smallest subgroup containingS, namely the intersection of all of subgroups containingS; it is denoted by⟨S⟩ and is called thesubgroup generated byS. An element ofG is in⟨S⟩ if and only if it is a finite product of elements ofS and their inverses, possibly repeated.[6]
Every elementa of a groupG generates a cyclic subgroup⟨a⟩. If⟨a⟩ isisomorphic to (the integersmodn) for some positive integern, thenn is the smallest positive integer for whichan =e, andn is called theorder ofa. If⟨a⟩ is isomorphic to thena is said to haveinfinite order.
The subgroups of any given group form acomplete lattice under inclusion, called thelattice of subgroups. (While theinfimum here is the usual set-theoretic intersection, thesupremum of a set of subgroups is the subgroupgenerated by the set-theoretic union of the subgroups, not the set-theoretic union itself.) Ife is the identity ofG, then the trivial group{e} is theminimum subgroup ofG, while themaximum subgroup is the groupG itself.
G is the group theintegers mod 8 under addition. The subgroupH contains only 0 and 4, and is isomorphic to There are four left cosets ofH:H itself,1 +H,2 +H, and3 +H (written using additive notation since this is anadditive group). Together they partition the entire groupG into equal-size, non-overlapping sets. The index[G :H] is 4.
Given a subgroupH and somea inG, we define theleftcosetaH = {ah :h inH}. Becausea is invertible, the mapφ :H →aH given byφ(h) =ah is abijection. Furthermore, every element ofG is contained in precisely one left coset ofH; the left cosets are the equivalence classes corresponding to theequivalence relationa1 ~a2if and only if is inH. The number of left cosets ofH is called theindex ofH inG and is denoted by[G :H].
where|G| and|H| denote theorders ofG andH, respectively. In particular, the order of every subgroup ofG (and the order of every element ofG) must be adivisor of|G|.[7][8]
Right cosets are defined analogously:Ha = {ha :h inH}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to[G :H].
IfaH =Ha for everya inG, thenH is said to be anormal 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, ifp is the lowest prime dividing the order of a finite groupG, then any subgroup of indexp (if such exists) is normal.
This group has two nontrivial subgroups:■J = {0, 4} and■H = {0, 4, 2, 6}, whereJ is also a subgroup ofH. The Cayley table forH is the top-left quadrant of the Cayley table forG; The Cayley table forJ is the top-left quadrant of the Cayley table forH. The groupG iscyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.[9]
S4 is thesymmetric group whose elements correspond to thepermutations of 4 elements. Below are all its subgroups, ordered by cardinality. Each group(except those of cardinality 1 and 2) is represented by itsCayley table.
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