A 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''.
- ), usually to emphasize the operation --- when ''G'' carries multiple algebraic or other structures.
- and writing the product ''a''---''b'' as simply ''ab''.
- ''H'' is a subgroup of the group ''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 Iff ''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 = ''a''''n'' − 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 ''e''''G'', and ''H'' is a subgroup of ''G'' with identity ''e''''H'', then ''e''''H'' = ''e''''G''.
- 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'' = ''e''''H'', then ''ab'' = ''ba'' = ''e''''G''.
- The intersection of subgroups ''A'' and ''B'' is again a subgroup. 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.
- 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 /''n'' for some positive integer ''n'', then ''n'' is the smallest positive integer for which ''a''''n'' = ''e'', and ''n'' is called the ''order'' of ''a''. If <''a''> is isomorphic to , then ''a'' is said to have ''infinite order''.
- The subgroups of any given group form a Complete Lattice under inclusion. (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.
Let ''G'' be the Abelian Group whose elements are
and whose group operation is Addition Modulo Eight . Its Cayley Table is
This group has a pair of 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.
Given a subgroup ''H'' and some ''a'' in G, we define the ''left states that
:
where o(''G'') and o(''H'') denote the Order s of ''G'' and ''H'', respectively. In particular, if ''G'' is finite, then the order of every subgroup of ''G'' (and the order of every element of ''G'') must be a divisor of o(''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 : ''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.
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