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{\bf \Large 250A Homework 2}\\[7mm]
{\small Due Monday October 18}\\[4mm]
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\noindent
{\bf Question 1} 
Let $G$ be a group and suppose $x^2=1$ $\forall x\in G$. Prove that
$G$ is abelian.

\vspace{.4cm}\noindent
{\bf Question 2} 
Let $G=\{x_1,\ldots, x_n\}$ be a finite abelian group.
Prove $$(x_1\cdots x_n)^2=1.$$

\vspace{.4cm}\noindent
{\bf Question 3} Find all groups of order 7 or less.
 
\vspace{.4cm}\noindent
{\bf Question 4} 
Let $\Real^+$ and $\Real^*$ be the group of real numbers
under addition and non-zero real numbers under multiplication, 
respectively. Show that these groups are not isomorphic.


\vspace{.4cm}\noindent
{\bf Question 5} 
Show that the $n$th roots of unity of 
$\{z\in {\mathbb C}|z^n=1,n\in{\mathbb N}\}$ form 
a group under multiplication but not addition.

\vspace{.4cm}\noindent
{\bf Question 6} 
Let $x\in G$ a group. Show that $|x|=n<\infty$ implies
that $1,x,x^2,\ldots,x^{n-1}$ are distinct.

\vspace{.4cm}\noindent
{\bf Question 7}
Suppose ${\cal N}$ is a nilpotent operator on a vector
space $V$. In this case $V$ is a direct sum of cyclic
subspaces. Show that the number of summands is $\dim\ker{\cal N}$.


 
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