# Orders of p-groups, normal subgroups

• Apr 11th 2010, 01:37 PM
kimberu
Orders of p-groups, normal subgroups
If H is a p-group show every element can be multiplied by some power to equal p.
• Apr 11th 2010, 02:22 PM
FancyMouse
a. No need to bring in Cauchy. For any x in G, if x is not the identity, then x has an order p^r for some r>0. What about $\displaystyle x^{p^{r-1}}$?
b. Further hint: prove that $\displaystyle x\in N$ and $\displaystyle x\in L$, using the fact that N,L are normal in G
c. Suppose A is the subgroup of N of order 2, B is the subgroup of L of order 3. We know that everything in N and L commutes. In particular, everything in A commutes with everything in B. But A and B are cyclic. What is the subgroup AB look like? (Notice that the "subgroup" AB makes sense because $\displaystyle A\cap B=\{e\}$ and everything in A commutes with everything in B)
• Apr 11th 2010, 02:40 PM
(b) is so intuitive to me, but the proof your supposed to find is so opaque. Can't you use that G is the semidirect product of N and L somehow?
• Apr 11th 2010, 03:25 PM
kimberu
Quote:

Originally Posted by FancyMouse
a. No need to bring in Cauchy. For any x in G, if x is not the identity, then x has an order p^r for some r>0. What about $\displaystyle x^{p^{r-1}}$?
b. Further hint: prove that $\displaystyle x\in N$ and $\displaystyle x\in L$, using the fact that N,L are normal in G
c. Suppose A is the subgroup of N of order 2, B is the subgroup of L of order 3. We know that everything in N and L commutes. In particular, everything in A commutes with everything in B. But A and B are cyclic. What is the subgroup AB look like? (Notice that the "subgroup" AB makes sense because $\displaystyle A\cap B=\{e\}$ and everything in A commutes with everything in B)

Thank you! I understand (c), and I think for (b), after showing x is in N and L, this implies that x = 1 = $\displaystyle aba^{-1}b^{-1} = b(aa^{-1})b^{-1}$ which implies ab = ba? (Or am I completely wrong?

But I'm still lost on part (a), could you possibly elaborate further? What is the significance of $\displaystyle x^{p^{r-1}}$?
• Apr 11th 2010, 05:06 PM
aliceinwonderland
Quote:

Originally Posted by kimberu
Thank you! I understand (c), and I think for (b), after showing x is in N and L, this implies that x = 1 = $\displaystyle aba^{-1}b^{-1} = b(aa^{-1})b^{-1}$ which implies ab = ba? (Or am I completely wrong?

But I'm still lost on part (a), could you possibly elaborate further? What is the significance of $\displaystyle x^{p^{r-1}}$?

Well, that is not correct.

$\displaystyle aba^{-1} \in L$ and $\displaystyle (aba^{-1})b^{-1} \in L$.
$\displaystyle ba^{-1}b^{-1} \in N$ and $\displaystyle a(ba^{-1}b^{-1}) \in N$.Then, $\displaystyle aba^{-1}b^{-1} \in N \cap L = \{e\}$.
Thus $\displaystyle ab=ba$ for all $\displaystyle a \in N, b \in L$.
• Apr 11th 2010, 05:21 PM
kimberu
Quote:

Originally Posted by aliceinwonderland
Well, that is not correct.

$\displaystyle aba^{-1} \in L$ and $\displaystyle (aba^{-1})b^{-1} \in L$.
$\displaystyle ba^{-1}b^{-1} \in N$ and $\displaystyle a(ba^{-1}b^{-1}) \in N$.Then, $\displaystyle aba^{-1}b^{-1} \in N \cap L = \{e\}$.
Thus $\displaystyle ab=ba$ for all $\displaystyle a \in N, b \in L$.

Sorry, I'm still unclear -- Why does $\displaystyle aba^{-1}b^{-1} \in N \cap L = \{e\}$ imply that ab =ba, if not for the reason I stated before?
• Apr 11th 2010, 05:26 PM
aliceinwonderland
Quote:

Originally Posted by kimberu
Sorry, I'm still unclear -- Why does $\displaystyle aba^{-1}b^{-1} \in N \cap L = \{e\}$ imply that ab =ba, if not for the reason I stated before?

$\displaystyle aba^{-1}b^{-1} = 1 \leftrightarrow aba^{-1}= b \leftrightarrow ab=ba$.