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For example, the subgroup Z 7 of the non-abelian group of order 21 is normal (see List of small non-abelian groups and Frobenius group#Examples). An alternative proof of the result that a subgroup of index lowest prime p is normal, and other properties of subgroups of prime index are given in .
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.
The lowest order for which the cycle graph does not uniquely represent a group is order 16. In the lists of subgroups, the trivial group and the group itself are not listed. Where there are several isomorphic subgroups, the number of such subgroups is indicated in parentheses. Angle brackets <relations> show the presentation of a group.
More generally, a subgroup, , of finite index, , in contains a subgroup, , normal in and of index dividing ! called the normal core. In particular, if p {\displaystyle p} is the smallest prime dividing the order of G {\displaystyle G} , then every subgroup of index p {\displaystyle p} is normal.
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words, H is a maximal element of the partially ordered set of subgroups of G that are not equal to G.
The generalized Fitting subgroup is the smallest subgroup that contains the Fitting subgroup and all normal semisimple subgroups. The generalized Fitting subgroup can also be viewed as a generalized centralizer of chief factors. A nonabelian semisimple group cannot centralize itself, but it does act on itself as inner automorphisms.
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.