# Template:Abelian normal subgroups of finite p-group facts to check against

From Groupprops

FACTS TO CHECK AGAINSTfor abelian normal subgroup of group of prime power order:EXISTENCE: existence of abelian normal subgroups of small prime power orderCONGRUENCE, IMPLIES ABELIAN-TO-NORMAL REPLACEMENT: congruence condition on number of subgroups of given prime power order (covers cases of and ) | congruence condition on number of abelian subgroups of prime-cube order | congruence condition on number of abelian subgroups of prime-fourth order | Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order | congruence condition on number of abelian subgroups of prime index | congruence condition on number of abelian subgroups of prime-square index for odd prime |abelian-to-normal replacement theorem for prime-square index | abelian-to-normal replacement theorem for prime-cube index for odd primeCONGRUENCE CONDITIONS BASED ON EXPONENT:PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]RELATION WITH REPRESENTATION THEORY: degree of irreducible representation divides index of abelian normal subgroup