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Welcome to Groupprops, The Group Properties Wiki (beta). 8000+ articles, including most basic group theory material. It is managed by Vipul Naik, a Ph.D. in Mathematics at the University of Chicago. It is part of a broader subject wikis initiative -- see the subject wikis reference guide for more details.

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<random>Permutable subgroup (DEFINITION): A subgroup that permutes with every other subgroup. May not be normal.@@@Hall subgroup (DEFINITION): A subgroup of a finite group whose order and index are relatively prime. Sylow subgroups are a special case.@@@Complete group (DEFINITION): A group whose center is trivial and for which every automorphism is inner.@@@Class-preserving automorphism: An automorphism of a group that sends every element to within its conjugacy class. May not be inner.@@@Baer norm (DEFINITION): The intersection of the normalizers of all subgroups of the group. Contains the center, and is a hereditarily normal subgroup.@@@Highly transitive group action (DEFINITION): A group action that is $k$-transitive for all natural numbers $k$.@@@Subnormal series: An ascending (resp., descending) series of subgroups with each subgroup normal in its successor (resp., predecessor).@@@Transitive subgroup property (DEFINITION): A subgroup satisfying the property inside a subgroup satisfying the property satisfies the property.</random>
<random>The union of all conjugates of a proper subgroup in a finite group can never be the whole group (FACT): And this breaks down for infinite groups.@@@The product of two proper conjugate subgroups of a group (not necessarily finite) is proper (FACT): However, the subgroup they generate could be the whole group.@@@Nilpotent implies center is normality-large (FACT): In a nilpotent group, the center intersects every nontrivial normal subgroup nontrivially.@@@Characteristic of normal implies normal (FACT): And characteristicity is also the tightest such property.@@@Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup (FACT): This, incidentally, helps prove that maximal among abelian normal implies self-centralizing in supersolvable.</random>
<random>Nilpotent versus solvable (SURVEY ARTICLE): There are important differences between a nilpotent group and a solvable group.@@@Characteristic versus normal (SURVEY ARTICLE): There are important differences between a characteristic subgroup and a normal subgroup.@@@Varying normality (SURVEY ARTICLE): Ever wondered how to take the subgroup property of being normal and create lots of variations?@@@Understanding the definition of a group (SURVEY ARTICLE): Explore the different aspects of the definition.@@@History of groups (SURVEY ARTICLE): Learn more about the history of groups.</random>

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