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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.
Since the normal subgroup is a subgroup of H, its index in G must be n times its index inside H. Its index in G must also correspond to a subgroup of the symmetric group S n, the group of permutations of n objects. So for example if n is 5, the index cannot be 15 even though this divides 5!, because there is no subgroup of order 15 in S 5.
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. [ 1 ] [ 2 ] Because every conjugation map is an inner automorphism , every characteristic subgroup is normal ; though the converse is not guaranteed.
The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient ...
A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group. [10] A concrete example of a normal subgroup is the subgroup = {(), (), ()} of the symmetric group, consisting of the identity and both three
If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup N G (H). If S is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains S is the subgroup C G (S). A subgroup H of a group G is called a self-normalizing subgroup of G if N G (H) = H.
A subgroup N of a group G is a normal subgroup of G if and only if for all elements g of G the corresponding left and right cosets are equal, that is, gN = Ng. This is the case for the subgroup H in the first example above. Furthermore, the cosets of N in G form a group called the quotient group or factor group G / N.
A significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory .