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{\Huge 189-235A: Basic Algebra I} \\ \vskip 0.1in
{\Huge Assignment 8} \\ \vskip 0.1in
{\Large Due: Wednesday, November 23 }
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1. Let $F$ be a field, and define a binary composition law on
$G=F-\{1\}$ by the rule
$$ a * b = a+b -ab.$$
Show that $G$, with this operation, is a group. (In particular,
write down the neutral element for $*$, and give a
formula for the inverse of $a\in G$.
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2. List all the elements of order $3$ in $S_3$. How many are there?
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3. List all the elements of order $6$ in $S_5$. How many such elements
are there?
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4. Given an example of non-abelian groups of order $12$ and $30$.
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5. Suppose that $G$ is a group in which $x^2=1$, for all $x\in G$. Show
that $G$ is abelian.
Give an example of a {\bf non-abelian } group $G$ of order $27$
in which $x^3=1$ for all $x\in G$.
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6. Show that the groups $S_3$ and ${\bf GL}_2(\Z_2)$ are isomorphic,
by writing down an isomorphism between them.
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7. A {\em transposition} in the symmetric group $S_n$ is a permutation of the
form $(ab)$ (i.e., a permutation that interchanges two elements
$a,b\in \{1,\ldots, n\}$, leaving all other $n-2$ elements fixed.
Show that every permutation in $S_n$
can be expressed as a product of transpositions.
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