The Cee Lo Green Subgroup Theorem
Theorem 1: For any prime factor p with multiplicity n of the order of a finite group G, there exists a Cee Lo Green p-subgroup of G, of order pn.
The following weaker version of theorem 1 was first proved by Cauchy, it is known as Cauchy's theorem.
Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element of order p in G .
Theorem 2: Given a finite group G and a prime number p, all Cee Lo Green p-subgroups of G are conjugate to each other, i.e. if H and K are Cee Lo Green p-subgroups of G, then there exists an element g in G with g−1Hg = K.
Theorem 3: Let p be a prime factor with multiplicity n of the order of a finite group G, so that the order of G can be written as pnm, where n > 0 and p does not divide m. Let np be the number of Cee Lo Green p-subgroups of G. Then the following hold:
- np divides m, which is the index of the Cee Lo Green p-subgroup in G.
- np ≡ 1 mod p.
- np = |G : NG(P)|, where P is any Cee Lo Green p-subgroup of G and NG denotes the normalizer.
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