# Properties of groups

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• Mar 5th 2009, 12:42 PM
iain3915
Properties of groups
• Mar 5th 2009, 04:13 PM
kalagota
3)
\$\displaystyle (a^m)^n = \underbrace{a^m+a^m+\cdots a^m}_{n \mbox{ times}} = \underbrace{\underbrace{a+a+\cdots a}_{m \mbox{ times}}+\underbrace{a+a+\cdots a}_{m \mbox{ times}}+\cdots \underbrace{a+a+\cdots a}_{m \mbox{ times}}}_{n \mbox{ times}}\$

\$\displaystyle = \underbrace{a+a+\cdots a}_{mn \mbox{ times}} = a^{mn} \$

6) this has been proven many times here.. but anyways,

let \$\displaystyle |ab|=n\$ and assume \$\displaystyle |ba|=d\$. show \$\displaystyle n=d\$.
suppose \$\displaystyle n>d\$.. \$\displaystyle (ba)^d = e \Rightarrow \underbrace{baba\cdots ba}_{d \mbox{ times}} = e\$

\$\displaystyle b^{-1}\underbrace{baba\cdots ba}_{d \mbox{ times}}\underbrace{baba\cdots ba}_{n-d \mbox{ times}}b = b^{-1}\underbrace{baba\cdots ba}_{n-d \mbox{ times}}b\$

\$\displaystyle e=\underbrace{aba\cdots bab}_{n \mbox{ times}}= \underbrace{aba\cdots bab}_{n-d \mbox{ times}}\$

and continue....

if \$\displaystyle n<d\$
\$\displaystyle \underbrace{baba\cdots ba}_{d \mbox{ times}} = e\$

\$\displaystyle \underbrace{ba\cdots b\underbrace{aba\cdots bab}_{n \mbox{ times}}a\cdots ba}_{d \mbox{ times}} = e\$

\$\displaystyle babae\cdots ba=\underbrace{baba\cdots ba}_{d-n \mbox{ times}} = e\$

....
• Mar 5th 2009, 04:31 PM
iain3915
Thanks for the reply.

I think the equation in 3 needs multiply signs though.
• Mar 5th 2009, 04:53 PM
kalagota
multiple signs? what the do you mean?
• Mar 5th 2009, 04:58 PM
iain3915
Because (a^m)^n does not equal a^m+a^m+a^m+a^m n times
• Mar 5th 2009, 04:59 PM
kalagota
ahh... yeah.. i mean, \$\displaystyle \underbrace{a^ma^ma^m\cdots a^m}_{n \mbox{ times}}\$... that should be it.. sorry for that..

and, all you have to do is to remove that + signs..

but anyways, if i assume that the operator is denoted by +, everything should be fine.. it is just a matter of how you denote your operator..
• Mar 5th 2009, 05:02 PM
iain3915
Do you know of an alternative way of doing these proofs without using all the n times sort of things?

It seems a bit ambiguous as it is.
• Mar 5th 2009, 05:05 PM
kalagota
it's the best proof and you will see that proof in many references.. (and i can't think of another for now..)

have you read my notes in the previous post? i edited it..