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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.
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In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. [ 1 ] [ 2 ] The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian .
A subgroup H of a group G is ascendant if there is an ascending subgroup series starting from H and ending at G, such that every term in the series is a normal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup is subnormal. automorphism An automorphism of a group is an isomorphism of the group to ...
Just as the upper central series and lower central series are extremal among central series, there are analogous series extremal among nilpotent series.. For a finite group H, the Fitting subgroup Fit(H) is the maximal normal nilpotent subgroup, while the minimal normal subgroup such that the quotient by it is nilpotent is γ ∞ (H), the intersection of the (finite) lower central series ...
In the mathematical field of group theory, a subgroup H of a given group G is a serial subgroup of G if there is a chain C of subgroups of G extending from H to G such that for consecutive subgroups X and Y in C, X is a normal subgroup of Y. [1] The relation is written H ser G or H is serial in G. [2]
By contrast, the subgroup of all rotations about the origin is not a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin.
Most groups of small order have a Sylow p subgroup P with a normal p-complement N for some prime p dividing the order, so can be classified in terms of the possible primes p, p-groups P, groups N, and actions of P on N. In some sense this reduces the classification of these groups to the classification of p-groups.