Main Page

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.

NEED HELP WITH UNDERGRADUATE LEVEL GROUP THEORY? If you want something specific, try the search bar! Else, try:
Basic definitions in group theory, basic facts in group theory, and elementary non-basic facts in group theory pages. There's much much more in the wiki!
Pages on symmetric group:S3 (see also subgroups, elements, representations), symmetric group:S4 (see also subgroups, elements, and representations), dihedral group:D8 (see also subgroups, elements, representations, and endomorphisms/automorphisms),symmetric group:S5 (see also subgroups, elements, and representations), quaternion group (see also subgroups, elements, and representations), alternating group:A4, alternating group:A5, and many more.
Incomplete (not fully finished) guided tour for beginners; the part prepared so far goes over the basic definitions of groups, subgroups, cosets, basic results such as Lagrange's theorem, and a little more, along with stimulating exercises.

<random>Suggested articles:
<random>Symmetric group:S3: The symmetric group of degree three (order six). Also, the dihedral group of degree three (order six). See also subgroups of S3, representations of S3, and elements of S3.@@@Dihedral group:D8 (also called dihedral group:D4): The dihedral group of degree four and order eight. See also subgroups of D8 and representations of D8.@@@Symmetric group:S4: The symmetric group of degree four (order 24). See also subgroups of S4 and representations of S4.</random>
<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>@@@Random suggested stuff:
<random>Symmetric group:S3: The symmetric group of degree three (order six). Also, the dihedral group of degree three (order six). See also subgroups of S3, representations of S3, and elements of S3.@@@Dihedral group:D8 (also called dihedral group:D4): The dihedral group of degree four and order eight. See also subgroups of D8 and representations of D8.@@@Symmetric group:S4: The symmetric group of degree four (order 24). See also subgroups of S4 and representations of S4.</random>
<random>Thompson's critical subgroup theorem (FACT): Every group of prime power order has a critical subgroup: a characteristic subgroup that is commutator-in-center, Frattini-in-center, and self-centralizing.@@@Thompson's replacement theorem (FACT): If $A$ is an abelian subgroup of maximum order in a $p$-group $P$ and $B$ is an abelian subgroup such that $A$ normalizes $B$ but $B$ does not normalize $A$, we can replace $A$ be another abelian subgroup $A^{*}$ of maximum order that normalizes $A$ with $A\cap B$ a proper subgroup of $A^{*}\cap B$.@@@Sylow's theorem with operators (FACT): An analogue of Sylow's theorem where, instead of looking at $p$-subgroups, we consider the $p$-subgroups invariant under the action of a coprime automorphism group.</random>
<random>Category:Subgroup properties: A listing of hundreds of properties that can be evaluated for a group and a subgroup thereof, such as normal subgroup, maximal subgroup, characteristic subgroup. Salient ones are at Category:Pivotal subgroup properties.@@@Category:Group properties: A listing of hundreds of properties that can be evaluated for a group, and are invariant up to isomorphism. Salient ones are at Category:Pivotal group properties.@@@Category:Group property implications: A listing of pages giving the statement and proof of group property implications: one group property implying another. For instance, abelian implies nilpotent.@@@Category:Subgroup property implications: A listing of pages giving the statement and proof of a property implication between subgroup properties. For instance, characteristic implies normal.@@@Category:Subgroup metaproperty satisfactions: A listing of pages proving that a subgroup property is well-behaved in some sense: it satisfies a subgroup metaproperty. For instance, normality is strongly intersection-closed and characteristicity is transitive.</random>
<random>Category:Characteristic subgroup-closed group properties: A list of group properties that are characteristic subgroup-closed: any characteristic subgroup of a group with the property also has the property.@@@Category:Applications of characteristic of normal implies normal: Some results that apply the fact that any characteristic subgroup of a normal subgroup is normal.@@@Category:Commutator computations: Facts about properties of the commutator of two subsets, or between a subgroup and a subset, in terms of the properties of the original subsets.@@@Category:Existence statements about arbitrarily large subnormal depth: Facts that guarantee the existence of subnormal subgroups of arbitrarily large subnormal depth satisfying certain additional conditions.@@@Category:Constraints on numerical invariants for finite groups: A list of constraints that can be placed on numerical invariants associated with finite groups.@@@Category:Congruence conditions: A list of the many statements in group theory that involve congruence conditions.@@@Category:Facts about odd-order groups that break down for even-order groups: A list of facts that are true for the finite odd-order groups but break down (in general) for groups of even order.@@@Category:Facts about groups of coprime order whose proof requires the assumption that one of them is solvable: These facts are true for all finite groups because coprime implies one is solvable, a consequence of the odd-order implies solvable.</random></random>