# Template:Weekly highlight

From Groupprops

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

Past weekly highlights

WEEKLY HIGHLIGHT: Pages on groups of a particular order. These pages provide summary and contrast information on all groups of a particular order. Check out, for instance: groups of order 8 (see also elements, representations), groups of order 16 (see also elements, representations), groups of order 32 (see also elements). For more large-scale trendspotting, check out groups of order 2^n