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{\bf Math 113 Homework 4, due 2/19/2019}
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{\bf \noindent\large Make sure you are using the 7th edition of Abstract algebra by Fraleigh -- if you do the wrong problems, you won't get points!}

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{\bf 1.} Book exercises 5.1-5.7

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{\bf 2.} Book exercises 5.11-5.13

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{\bf 3.} Book exercises 5.21, 5.27, 5.28

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{\bf 4.} Book exercise 4.28.

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{\bf 5.} {\bf (a)} Write down an addition table for the Weyl 4-group $V$ (look it up in the book!). Write down an addition table for the Gaussian numbers modulo $2$, i.e. $\G/2\cdot \G$. Give a function $V\to \G/2\cdot \G$ which takes one table to the other (i.e. is an isomorphism).\footnote{Recall that Gaussian numbers are numbers of the form $a+bi$ with $a,b$ integers. The group $\G/n\cdot G$ is the group of residues $\G/\sim$ where $a+bi\sim a'+b'i$ if the difference is $n\cdot k$ for $k = k_1+ k_2\cdot i$ a Gaussian number.}

{\bf (b)} Recall that the direct product $\Z_n\times \Z_n$ is the group of pairs $([a], [b])$ of residues modulo $n$ with \emph{componentwise} addition $([a], [b]) + ([a'], [b']) = ([a+a'], [b+b']).$ Construct an isomorphism from $\Z_n\times \Z_n$ to $(\G/n\cdot \G, +)$. 

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{\bf 6.} Extra credit, worth either 1/2 a problem or alternatively write ``doing this problem instead of problem x'' to replace one of problems 1.-4.\ but not 5: {\bf book exercise 4.29}


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