How Can Finite Group Theory Problems Be Solved?

[hgj]

Group Theory, please help!

Okay, so I'm stuck on a couple questions from my homework, and any guidance would be much appreciated.

1. Prove that if G is a finite group with an even number of elements,
then there is an element x in G such that x is not the identity and
x^2 = e.

I know there exists some element x in G because G is not empty. And because e (the identity element in G) is unique, x is not equal to e, so x is not the identity. But I can't see how to go from x doesn't equal e to x^2 = e.

2. Prove that if (S,*) is a finite set with a binary operation that is
associative, has an identity, and satisfies the cancellation laws,
then (S,*) is a group.

I know that for (S, *) to be a group, it must be associative, there must exist an identity element e in S wrt *, and every element in S must be invertible. The first two properties follow easily from the way (S,*) is defined, but I don't know how to show the last property holds. It makes sense to me that it's true when I look at the cancellation law (if a,b,c are in S and ab = ac, the b = c), and I've tried working backwards, but then I find myself wanting to create an inverse in S, and that seems wrong.

 

[AKG]

When x² = e, then x = x^-1. However, if x² is not equal to e, then x and x^-1 are distinct. If the order of G is even, then think about what happens if x and x^-1 form distinct pairs for each non-identity x.

 

[Muzza]

2. Prove that if (S,*) is a finite set with a binary operation that is
associative, has an identity, and satisfies the cancellation laws,
then (S,*) is a group.

Suppose the elements of S are x_1, \ldots, x_n. Let x_k be some arbitrary element. What happens when you consider the elements

x_1 x_k, x_2 x_k, \cdots, x_n x_k?

Can two of them be equal to each other, for example?

 

[hgj]

Suppose the elements of S are x_1, \ldots, x_n. Let x_k be some arbitrary element. What happens when you consider the elements

x_1 x_k, x_2 x_k, \cdots, x_n x_k?

Can two of them be equal to each other, for example?

I see if i let x_1x_k = x_2x_k = \cdots = x_nx_k then, through the cancellation law, I get x_1 = x_2 = \cdots = x_n. Then, because it is given that S has an identity element, if these are all the elements of S and they are all equal, then they must all equal that identity element. So, each element in S has an inverse. But how can I claim x_1x_k = x_2x_k = \cdots = x_nx_k ?

 

[AKG]

You don't need all of them to be equal. But can any pair be equal? If any pair at all is equal, then there is an i and a j distinct from each other such that x_ix_k = x_jx_k and by the right cancellation law, x_i = x_j. But this contradicts the stipulation that x_i and x_j are distinct if i and j are distinct. So it cannot the the case that x_ix_k = x_jx_k for any i and j. If S has n distinct elements, and x_1 x_k, x_2 x_k, \cdots, x_n x_k are n distinct elements of S, then clearly S = \{x_1 x_k, x_2 x_k, \cdots, x_n x_k\}. Since identity is an element of S, there is some, let's call it m, such that x_mx_k = e. This should be more than enough, you can tie up the loose ends to complete the proof.