I have two questions, one of which I am pretty sure I have answered, but I would like your opinion on it.
1) Let G be a finite group of order 12, is it possible that the center of this group has order 4?
I am not even sure how to approach this. I did have to look up the definition of the center of a group and I know that the center of a group G is defined as:
2) Suppose that the order of some finite Abelian group G is divisible by 42. Prove that G has a cyclic subgroup of order 42.
Let G be a finite abelian group of order 42. Remember that the Fundamental Theorem of Finite Abelian Groups states that any finite abelian group can be written as where are not necessarily distinct and where . And a corollary follows stating that if divides the order of a finite abelian group , then has a subgroup of order .
Now since 42 divides the order of G, and G is abelian then we know that G contains a subgroup of order 42. Let H be such a subgroup. Since G is abelian, H is abelian. Since has order , we see that . Since is cyclic for any prime , is cyclic.
Does that work?