# Elementary abelian group

• Aug 24th 2011, 10:25 AM
abhishekkgp
Elementary abelian group
Find the number of subgroups of index $\displaystyle p$ of the elementary abelian group $\displaystyle E_{p^n}$.
The answer should be $\displaystyle \frac{p^n-1}{p-1}$ but i don't see how to approach this problem.
• Aug 24th 2011, 04:25 PM
Drexel28
Re: Elementary abelian group
Quote:

Originally Posted by abhishekkgp
Find the number of subgroups of index $\displaystyle p$ of the elementary abelian group $\displaystyle E_{p^n}$.
The answer should be $\displaystyle \frac{p^n-1}{p-1}$ but i don't see how to approach this problem.

Note first that $\displaystyle E_{p^n}$ is a vector space over $\displaystyle \mathbb{F}_p$. Moreover, if $\displaystyle H\leqslant E_{p^n}$ then it's also trivially a subspace of $\displaystyle E_{p^n}$ since it's closed under addition. Thus, the subgroups of index $\displaystyle p$, which are precisely the subgroups of order $\displaystyle p^{n-1}$, which are precisely the subspaces of dimension $\displaystyle n-1$. Now, using the common formula that the number of subspaces of an $\displaystyle n$-dimensional $\displaystyle \mathbb{F}_q$-space is $\displaystyle \displaystyle \frac{(q^n-1)(q^n-q)\cdots(q^n-q^{k-1})}{(q^k-1)(q^{k}-q)\cdots(q^k-q^{k-1})}$ (which is just a simple counting argument about choosing linearly independent subsets of size $\displaystyle k$--it's on page 412 of Dummit and Foote for example) and pluggin in $\displaystyle k=n-1$ gives the desired result.
• Aug 24th 2011, 07:06 PM
abhishekkgp
Re: Elementary abelian group
Quote:

Originally Posted by Drexel28
Note first that $\displaystyle E_{p^n}$ is a vector space over $\displaystyle \mathbb{F}_p$. Moreover, if $\displaystyle H\leqslant E_{p^n}$ then it's also trivially a subspace of $\displaystyle E_{p^n}$ since it's closed under addition. Thus, the subgroups of index $\displaystyle p$, which are precisely the subgroups of order $\displaystyle p^{n-1}$, which are precisely the subspaces of dimension $\displaystyle n-1$. Now, using the common formula that the number of subspaces of an $\displaystyle n$-dimensional $\displaystyle \mathbb{F}_q$-space is $\displaystyle \displaystyle \frac{(q^n-1)(q^n-q)\cdots(q^n-q^{k-1})}{(q^k-1)(q^{k}-q)\cdots(q^k-q^{k-1})}$ (which is just a simple counting argument about choosing linearly independent subsets of size $\displaystyle k$--it's on page 412 of Dummit and Foote for example) and pluggin in $\displaystyle k=n-1$ gives the desired result.

thank you drexel but your solution is too advanced for me... actually i wanted to solve the question given on pg 168(question 8(b)) of dummit and foote. i had posted the question on this thread http://www.mathhelpforum.com/math-he...reply&t=186533 . then i took a special case of this problem which is "prove that number of subgroups of order $\displaystyle p$ in $\displaystyle E_{p^n}$ is equal to the number of subgroups of index $\displaystyle p$ in $\displaystyle E_{p^n}$." Now i was able to prove that $\displaystyle E_{p^n}$ has $\displaystyle \frac{p^n-1}{p-1}$ subgroups of order $\displaystyle p$.

I want to solve the question using only the material covered till pg. 168. Your solution uses Vector spaces which is given on pg. 388 and i have not yet read that.. :(
• Aug 24th 2011, 07:59 PM
Drexel28
Re: Elementary abelian group
Quote:

Originally Posted by abhishekkgp
thank you drexel but your solution is too advanced for me... actually i wanted to solve the question given on pg 168(question 8(b)) of dummit and foote. i had posted the question on this thread http://www.mathhelpforum.com/math-he...reply&t=186533 . then i took a special case of this problem which is "prove that number of subgroups of order $\displaystyle p$ in $\displaystyle E_{p^n}$ is equal to the number of subgroups of index $\displaystyle p$ in $\displaystyle E_{p^n}$." Now i was able to prove that $\displaystyle E_{p^n}$ has $\displaystyle \frac{p^n-1}{p-1}$ subgroups of order $\displaystyle p$.

I want to solve the question using only the material covered till pg. 168. Your solution uses Vector spaces which is given on pg. 388 and i have not yet read that.. :(

Use the fourth isomorphism theorem to reduce the problem to the case of elementary abelian groups and then take a look at my solution. I mean, not to be pushy, but the only alternative I can think of is a nasty argument using the Orbit Stabilizer theorem. It's easier to just use what I said, and realize that math isn't an insular subject--subjects are supposed to be intermingled.
• Aug 24th 2011, 08:06 PM
Drexel28
Re: Elementary abelian group
If you're unwaivering in your desire to do it 'by the book' a LONG solution can be found here.
• Aug 25th 2011, 10:04 AM
abhishekkgp
Re: Elementary abelian group
Quote:

Originally Posted by Drexel28
Use the fourth isomorphism theorem to reduce the problem to the case of elementary abelian groups.

I am not sure how to do that. Can you drop a hint?
• Aug 25th 2011, 12:34 PM
Drexel28
Re: Elementary abelian group
Quote:

Originally Posted by abhishekkgp
I am not sure how to do that. Can you drop a hint?

In the link I provided the guy does the same thing I would do. Try reading it.