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Define ~ S4 as follows: for f, g element of S4 f~g if and only if f(4) = g(4) this is easily seen to be an equivalence relation on S4 (you don't have to show this) let X = S4/ ~ be the set of all equivalence classes under ~. Define * X as [f] * [g] = [f o g]. Is * a well-defined operation on X? If so, prove this fact; if not explain why not.

2) Show that G is a group and N is a normal subgroup of G, that G/N is abelian if and only if aba^-1b^-1 exists in N for all a,b element of G

New to abstract algebra and trying to teach myself. Any help with these two problems would be greatly appreciated!

user111230
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1 Answers1

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First of all, '$\in$' is read as 'element of'.

1) No, it is not well-defined on $S4/\sim$. Find an $f$ for $g_1:=$ constant $4$ and $g_2:=$ identity ($x\mapsto x$) functions such that $\ g_1\circ f\not\sim g_2\circ f$.

2) Rather straightforward. The equivalence relation determined by $N$ is $x\sim y$ iff $xy^{-1}\in N$. If you haven't checked yet, verify that $\sim$ is indeed an equivalence relation, and --provided $N$ is a normal subgroup-- that it preserves group operation, i.e. the group operation in the quotient set $G/\sim\,$ is well defined.

And the exercise itself is then obvious: $$aba^{-1}b^{-1}=(ab)(ba)^{-1}\in N \iff ab\sim ba \iff [ab]=[ba] \iff [a][b]=[b][a]\,.$$

Berci
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