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a and b are conjugate in G G is a group; a, b G . For some x G , xax1 b . Conjugacy class of a G is a group; a G . The set cl (a) {xax1 | x G} . The number of Conjugates of a Let G be a finite group and let a be an element of G. Then, | cl (a)|| G : C(a)| . The Class Equation For any finite group G, | G | | G : C(a)| where the sum runs over one element a from each conjugacy class of G. p-group A group of order p n , where p is prime. p-Groups Have Nontrivial Centers Let G be a finite group whose order is a power of a prime p. Then Z (G) has more than one element. 2 Groups of Order p are Abelian If | G | p2 , where p is prime, then G is Abelian. Existence of Subgroups of Prime-Power Order (Sylow’s First Theorem) Let G be a finite group and let p be a prime. If p k divides | G | , then G has at least one subgroup of order pk . Sylow p-Subgroup G is a group and p is a prime divisor of | G | . p k divides | G | but p k 1 does not. A subgroup of order p k . Cauchy’s Theorem Let G be a finite group and p a prime that divides the order of G. Then G has an element of order p. Conjugate Subgroups Two subgroups H and K of a group G that are related by H gKg 1 , where g G . Sylow’s Second Theorem If H is a subgroup of a finite group G and | H | is a power of a prime p, the H is contained in some Sylow p-subgroup of G. Sylow’s Third Theorem The number of Sylow p-subgroups of G is equal to 1 modulo p and divides | G | . Furthermore, any two Sylow p-subgroups are conjugate. A Unique Sylow p-Subgroup Is Normal A Sylow p-subgroup of a finite group G is a normal subgroup of G if and only if it is the only Sylow p-subgroup of G. Cyclic Groups of Order pq If G is a group of order pq, where p and q are prime, p q , and p does not divide q 1, then G is cyclic. In particular, G is isomorphic to Z pq .