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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.
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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.
WEEKLY HIGHLIGHT: Under various conditions, a universal power map being an automorphism or an endomorphism can give information about a group being abelian. See abelian implies universal power map is endomorphism, inverse map is automorphism iff abelian, square map is endomorphism iff abelian, cube map is surjective endomorphism implies abelian, cube map is endomorphism iff abelian (if order is not a multiple of 3), nth power map is endomorphism implies every nth power and (n-1)th power commute, nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center
On the other hand, there are some universal power maps that give endomorphisms and automorphisms without the group being abelian: Frattini-in-center odd-order p-group implies (kp plus 1)-power map is automorphism, Frattini-in-center odd-order p-group implies p-power map is endomorphism