We can determine exactly which positive numbers occur as the number of Sylow p-subgroups of finite groups, at least modulo some detailed information about simple groups. What can we say about the number of subgroups of order p?
Monthly Archives: January 2011
Is Sylow’s theorem sharp?
Sylow’s theorem says the number of Sylow p-subgroups is congruent to 1 mod p. Does the converse hold? If n is a number congruent to 1 mod p, then is there a finite group with n Sylow p-subgroups?