Results 1 to 2 of 2

Thread: show even and odd

  1. #1
    Member
    Joined
    Feb 2008
    Posts
    125

    Question show even and odd

    Let G be a finite group
    a) Show that the number of elements x of G such that x^3=e is odd
    b) Show that the number of elements x of G such that x^2 does not equal e is even
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by mandy123 View Post
    Let G be a finite group
    a) Show that the number of elements x of G such that x^3=e is odd
    Let $\displaystyle X = \{ x\in G | x^3 = e\}$. Now if $\displaystyle x\not = e$ is in $\displaystyle X$ then $\displaystyle x^3 = e$ so $\displaystyle x$ has order $\displaystyle 3$. Therefore, $\displaystyle x\not = x^2$. But $\displaystyle (x^2)^3 = (x^3)^2 = e$. Thus, $\displaystyle x^2 \in X$. Let $\displaystyle X = \{ e,x_1,x_2,....,x_n,x_{n+1} \}$. Write them in such a way so that $\displaystyle x_1 = x_2^2,....,x_n=x_{n+1}^2$. Therefore, all $\displaystyle x_i$ can be paired with $\displaystyle x_{i+1}$. Thus, there are an even number (zero is allowed) of elements which have order three. But we also include $\displaystyle e$ and this makes the total number of elements odd.

    b) Show that the number of elements x of G such that x^2 does not equal e is even
    Similar to above.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Using the MVT to show this
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Nov 25th 2010, 11:19 AM
  2. Show the sum
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Oct 21st 2010, 03:28 AM
  3. Show that f o g is one-to-one
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jan 24th 2010, 06:29 PM
  4. Show that ln5 < 1 + 1/2 + 1/3 + 1/4.
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 6th 2009, 01:29 AM
  5. how to show show this proof using MAX
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jan 14th 2009, 12:05 PM

Search Tags


/mathhelpforum @mathhelpforum