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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.
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.
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
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.
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.
derived subgroup Synonym for commutator subgroup. direct product The direct product of two groups G and H, denoted G × H, is the cartesian product of the underlying sets of G and H, equipped with a component-wise defined binary operation (g 1, h 1) · (g 2, h 2) = (g 1 ⋅ g 2, h 1 ⋅ h 2).
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 ...
In mathematics, specifically group theory, a subgroup series of a group is a chain of subgroups: = = where is the trivial subgroup.Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups.