Results 1 to 4 of 4

Math Help - A Couple Questions Dealing With Finite Groups

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    152

    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).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Oct 2008
    From
    Guernsey
    Posts
    69
    1) The order of an element divides the order of the group. This means that there can be no "self-inverse" elements, that is x^2 = e (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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2008
    Posts
    152
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by Janu42 View Post
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. finite abelian groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 2nd 2010, 08:00 PM
  2. Quotient Groups - Infinite Groups, finite orders
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: August 11th 2010, 07:07 AM
  3. more about finite groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 27th 2009, 10:50 PM
  4. result about finite groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 10th 2009, 06:28 PM
  5. Finite groups of even cardinality
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: February 26th 2006, 10:49 AM

Search Tags


/mathhelpforum @mathhelpforum