<s>[INST] $(12) \notin \langle (132), (123456) \rangle$ How can I prove/disprove that as an element of $S_6, $ $(12) \notin \langle (132), (123456) \rangle$. We can use GAP to check this, but by hand it looks not so obvious. 
Let $r=(123456)$ is odd and $s=(132)$ is even , so any word in {$r^{\pm},{s^{\pm}}$}, can give an odd permutation as well as even. 
 [/INST] By hand:
$$(1\ 2)=(1\ 2\ 3\ 4\ 5\ 6)^4(1\ 3\ 2)(1\ 2\ 3\ 4\ 5\ 6)^4(1\ 3\ 2)(1\ 2\ 3\ 4\ 5\ 6)^5$$
This is how I worked it out:
$$(1\ 2\ 3\ 4\ 5\ 6)=(1\ 2)(2\ 3\ 4\ 5\ 6)$$
so
$$(1\ 2)=(1\ 2\ 3\ 4\ 5\ 6)(2\ 3\ 4\ 5\ 6)^{-1}=(1\ 2\ 3\ 4\ 5\ 6)[(2\ 3\ 4)(4\ 5\ 6)]^{-1}$$
and of course
$$(2\ 3\ 4)=(1\ 2\ 3\ 4\ 5\ 6)(1\ 2\ 3)(1\ 2\ 3\ 4\ 5\ 6)^{-1}=(1\ 2\ 3\ 4\ 5\ 6)(1\ 3\ 2)^{-1}(1\ 2\ 3\ 4\ 5\ 6)^{-1}$$
and
$$(4\ 5\ 6)=(1\ 2\ 3\ 4\ 5\ 6)^3(1\ 2\ 3)(1\ 2\ 3\ 4\ 5\ 6)^{-3}=(1\ 2\ 3\ 4\ 5\ 6)^3(1\ 3\ 2)^{-1}(1\ 2\ 3\ 4\ 5\ 6)^{-3}$$
etc.
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