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Welcome to Groupprops, The Group Properties Wiki (beta). 7000+ 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 stuff:
Symmetric group:S4: The symmetric group of degree four (order 24). See also subgroups of S4 and representations of S4.
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.
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:Applications of characteristic of normal implies normal: Some results that apply the fact that any characteristic subgroup of a normal subgroup is normal.

What we are: Eventually, a complete and reliable reference for group theory. For now, an exciting place to read definitions and facts of group theory, and navigate the relationships between them.