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Subgroup analysis refers to repeating the analysis of a study within subgroups of subjects defined by a subgrouping variable. For example: ...
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e} ). [2] [3] If H is a subgroup of G, then G is sometimes called an overgroup of H.
A strictly characteristic subgroup, or a distinguished subgroup, is one which is invariant under surjective endomorphisms.For finite groups, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being strictly characteristic is equivalent to characteristic.
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. A subgroup series is used in the subgroup method. Subgroup series are a special example of the use of filtrations in abstract algebra.
The Puig subgroup L(G) is the intersection of the subgroups L n for n odd, and the subgroup L * (G) is the union of the subgroups L n for n even. Properties [ edit ]
The blue and orange subgroups are clades; each shows its common ancestor stem at the bottom of the subgroup branch. The green subgroup alone does not count as a clade; it is a paraphyletic group with respect to the blue group, because it excludes the blue branch which has descended from the same common ancestor. The green and blue subgroups ...
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.
For any subgroup of , the following conditions are equivalent to being a normal subgroup of .Therefore, any one of them may be taken as the definition. The image of conjugation of by any element of is a subset of , [4] i.e., for all .