CINTA作业11-证明题

CINTA

1、For all $a, b\in G$, prove that $a b^n a^{-1} = (aba^{-1})^n$ for n in $Z$.
Proof:
$ab^na^{-1}=a*b*b*...*b*a^{-1}(中间有n个b)\\ \qquad\quad=a*b*(a^{-1}*a)*b*(a^{-1}*a)*...*b*(a^{-1}*a)*b*a^{-1}(补n-1个a^{-1}*a)\\ \qquad\quad=（a*b*a^{-1}）*（a*b*a^{-1}）*...(a*b*a^{-1})*(a*b*a^{-1})(结合律)\\ \qquad\quad=(aba^{-1})^n$

2、For all $a, b\in G$, $(ab)^2 = a^2 b^2$, prove that the group $G$ is abelian.
Proof:
$\forall a,b \in G$
$(ab)^2=a^2b^2$
$abab=aabb$
$a^{-1}(abab)b^{-1}=a^{-1}(aabb)b^{-1}$
$(a^{-1}a)ba(bb{-1})=(a^{-1}a)ab(bb^{-1})$
$ab=ba$
the group G is abelian.

3.Prove that the inverse of $g_0 g_1 .. g_n = g_n^{-1} g_{n-1}^{-1} ... g_0^{-1}.$
Proof:
$(g_0 g_1 .. g_n)( g_n^{-1} g_{n-1}^{-1} ... g_0^{-1})=$
$(g_0 (g_1 .. (g_n g_n^{-1}) g_{n-1}^{-1} ... g_1^{-1})g_0^{-1})=$
$ee .. .ee=$
$e$
$the\quad inverse\quad of\quad g_0 g_1 .. g_n = g_n^{-1} g_{n-1}^{-1} ... g_0^{-1}.$