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\begin{document}
\noindent {\bf Physics 251} \hfill {\bf Problem Set 1} \hfill {\bf
Spring 2019}
\myrule
\vskip 0.25in
\centerline{\sl DUE: \ Thursday April 18, 2019}
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\noindent
1. Consider the set $\mathbb{R}^2$ consisting of pairs of real numbers.
For $(x,y)\in \mathbb{R}^2$, define scalar multiplication by:
$c(x,y)=(cx,cy)$ for any real number $c$, and define vector
addition and multiplication as
follows:
\beqa
(x_1,y_1)+(x_2,y_2)&=&\,(x_1+x_2,y_1+y_2)\,,\label{e1} \\
(x_1,y_1)\cdot (x_2,y_2)&=&\,(x_1 x_2, y_1 y_2)\,.\label{e2}
\eeqa
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(a) Is $\mathbb{R}^2$ a group?
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(b) Is $\mathbb{R}^2$ a field?
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(c) Is $\mathbb{R}^2$ a linear vector space (over $\mathbb{R}$)?
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(d) Is $\mathbb{R}^2$ a linear algebra (over $\mathbb{R}$)?
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\noindent
Suppose that the multiplication law given by \eq{e2} is replaced by 
\beq
(x_1,y_1)\cdot (x_2,y_2)=\,(x_1 x_2-y_1 y_2\,,\,x_1 y_2+x_2 y_1)\,.\label{e3}
\eeq
Do any of the results obtained in parts (a)--(d) above change?  Identify a well know mathematical object that is isomorphic to $\mathbb{R}^2$ if \eq{e2} is replaced by \eq{e3}.


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%\noindent
%2. Consider the following two groups:
%\vskip 0.1in
%
%$T=\bigl\{\text{proper rotations that map a regular tetrahdron into itself}\bigr\}$\,,
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%$T_d=\bigl\{\text{proper rotations and reflections that map a regular tetrahdron into itself}\bigr\}$\,.
%\vskip 0.1in

%\noindent
%Show that the following isomorphisms are valid: $T\iso A_4$ and $T_d\iso S_4$.
\vskip 0.5in

\noindent
2. Consider the possibility that a set $G$ of $n\times n$ matrices forms a group with
respect to matrix multiplication.
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(a) Prove that if $G$ is a group and if one of the elements of $G$ is a non-singular matrix
then all of the elements of $G$ must be non-singular matrices.  Conclude that all
the elements of $G$ are either non-singular matrices or singular matrices.
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(b) Consider the set of $2\times 2$ singular matrices $G$ of the form
\beq \label{xxxx}
\begin{pmatrix} x &\,\,\,x \\ x & \,\,\, x\end{pmatrix}\,,
\eeq
where $x\in\mathbb{R}$ and $x\neq 0$.  Prove that $G$ is a group with respect to matrix multiplication.
Determine the matrix corresponding to the identity element of $G$. Determine the inverse of the 
element specified in~\eq{xxxx}.
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(c) The group defined in part (b) is isomorphic to a well known group.  Identify this group.


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\clearpage
\noindent
3. Consider the dihedral group $D_4$.
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(a) Write down the group multiplication table.
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(b) Enumerate the subgroups, the normal subgroups and the conjugacy classes.
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(c) Identify the factor groups.  Is the full group the direct product
of some of its subgroups?
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%(d) Write out the conjugacy class multiplication table.
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%
%(e) Determine explicitly the matrices of the regular representation.
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%(f) Write out an explicit irreducible two-dimensional representation
%of $D_4$.  Check that the group multiplication table is
%preserved.  Verify that this representation is irreducible.
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%(g) Construct the character table for the irreducible representations of $D_4$.


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\noindent
4. The {\it center} of a group $G$, denoted by $Z(G)$, is defined
as the set of elements $z\in G$ that commute with all elements
of the group.  That is,
$$Z(G)=\{z\in G\,|\,zg=gz\,,\forall\,g\in G\}\,.$$
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(a) Show that $Z(G)$ is an abelian subgroup of $G$.
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(b) Show that $Z(G)$ is a normal subgroup of $G$.
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(c) Find the center of $D_4$ and construct the group $D_4/Z(D_4)$.
Determine whether the isomorphism
$D_4\iso [D_4/Z(D_4)]\otimes Z(D_4)\,$ is valid.
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\noindent
5. An automorphism is defined as an isomorphism of a group $G$ onto
itself.
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(a) Show that for any $g\in G$, the mapping $T_g(x)=gxg^{-1}$ is
an automorphism (called an {\it inner automorphism}), where $x\in G$.
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(b) Show that the set of all inner automorphisms of $G$, denoted
by ${\cal I}(G)$, is a group.
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(c) Show that ${\cal I}(G)\simeq G/Z(G)$, where $Z(G)$ is the center
of $G$.
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(d) Show that the set of all automorphisms of $G$, denoted by
${\cal A}(G)$, is a group and that ${\cal I}(G)$ is an invariant
subgroup. (The factor group ${\cal A}(G)/{\cal I}(G)$ is called
the group of {\it outer automorphisms} of $G$.)
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%(e) Illustrate these results for $G=S_3$ and  $G=\mathbb{Z}$.
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\noindent
6. Consider an arbitrary orthogonal matrix $R$, which satisfies $RR^{\T}=\mathds{1}$
(where $\mathds{1}$ is the identity matrix).  
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(a) Prove that the possible values of ${\rm det}~\!R$ are $\pm 1$. [{\sl HINT:} Consider ${\rm det}(RR^{\T})$
and use one of the well-known properties of determinants.]
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(b) The group SO(2) consists of all $2\times 2$ orthogonal matrices
with unit determinant.  Prove that SO(2) is an abelian group. 
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(c) The group O(2) consists of all $2\times 2$ orthogonal
matrices, with no restriction on the sign of its determinant.  Is
O(2) abelian or non-abelian?  (If the latter, exhibit two O(2) matrices that do
not commute.)


 
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