Thread: A Couple Questions Dealing With Finite Groups

1) Let G be an abelian group such that the order of G is an odd integer. Show that the product of all the elements in G is e.

2) Show that the multiplication in Zp - {0} is associative. (Zp is the group of integers relatively prime to p (a prime) under multiplication modulo p).

1) The order of an element divides the order of the group. This means that there can be no "self-inverse" elements, that is (other than e of course)

Therefore every element has an inverse distinct from itself. Therefore by multiplying all elements together, we can pair off elements with their inverses, to get e

2) Use the fact that multiplication of integers is associative

I have one more:

a) Show that if p is prime then (p-1) is congruent to -1 (mod p). And it says hint: "Consider which elements of (Zp - {0}, mult. mod n) are their own inverses.

b) Prove the converse. Show that if n is greater than or equal to 2 and (n-1)! is congruent to -1 (mod n) then n is prime.

Originally Posted by Janu42

a) Show that if p is prime then (p-1) is congruent to -1 (mod p). And it says hint: "Consider which elements of (Zp - {0}, mult. mod n) are their own inverses.

Pair all the inverses together, so that they cancel to 1. The only inverses that have no pairs are +1 and -1, so you leave these alone how they are. And so when you take the product mod p you find that it simplifies to (1)(-1) = -1 (mod p).

b) Prove the converse. Show that if n is greater than or equal to 2 and (n-1)! is congruent to -1 (mod n) then n is prime.

See this.