- #1
kewljcs
- 7
- 0
1. If G is a finite group that does not contain a subgroup isomorphic to Z_p X Z_p for any prime p. prove that G is cyclic
im stumped. i don't understand the 'does not contain a subgroup isomorphoc to Z_p X Z_p part.
ive tried using cauchy's theorem for abelian group: if G is a finite abelian group, and let p be a prime that divides the order of G. then g has an element of order p.
so, Z_p is a finite cyclic gruop, but Z_p X Z_p is not because p, p are not relatively prime.
what else.. :/
2 question: Let G be a group of odd order. prove that the mapping x --> x^2 from G to itself is one to one.
i know that the smallest group of odd order is {I, a, a^-1}. i don't understand the mapping, basically.
im stumped. i don't understand the 'does not contain a subgroup isomorphoc to Z_p X Z_p part.
ive tried using cauchy's theorem for abelian group: if G is a finite abelian group, and let p be a prime that divides the order of G. then g has an element of order p.
so, Z_p is a finite cyclic gruop, but Z_p X Z_p is not because p, p are not relatively prime.
what else.. :/
2 question: Let G be a group of odd order. prove that the mapping x --> x^2 from G to itself is one to one.
i know that the smallest group of odd order is {I, a, a^-1}. i don't understand the mapping, basically.