If two groups have a solvable word, then their free product and direct product have a solvable word. I am having a rough time with this question. If there are any suggestions on how to start this, that would be very helpful.
Solvable word for the free product and direct product of two groups.
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group-theory
1 Answers
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To tell whether a word $w((g_1,h_1),\ldots,(g_n,h_n))$ in $G\times H$ is trivial, note that $w\bigl((g_1,h_1),\ldots,(g_n,h_n)\bigr) = \bigl( w(g_1,\ldots,g_n),w(h_1,\ldots,h_n)\bigr),$ and that $(x,y)\in G\times H$ is trivial if and only if $x$ is trivial and $y$ is trivial.
To tell whether a word in the free product $G*H$ is trivial, write it as a product $g_1h_1g_2h_2\cdots g_mh_m$ where $g_i\in G$, $h_i\in H$, $h_i\neq 1$ for $i=1,\ldots m-1$, and $g_i\neq 1$ for $i=2,\ldots,m$. This will be trivial if and only if $m=1$, $g_1=1$ and $h_1=1$.
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1@ Arturo, thanks so much. How do you do that so fast? – 2012-02-23