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{{\large \bf Incorporating the Software GAP into Teaching\\ Abstract Algebra
\\PREP Workshop 2006
\\ Project: Keeler and Swain
}}
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\noindent
The following series of exercises will explore the conjugacy classes for groups
and relationships between their sizes and the order of the group. First we will introduce
the useful commands in \textsf{GAP}. We will start by exploring the Group $A_4$. For any element
{\tt a} in a group \texttt{G}, the command \texttt{ConjugacyClass(G,a);} creates the
conjugacy class of \texttt{G} containing \texttt{a}. The command \\
\texttt{ConjugacyClasses(G);} creates a list of all the conjugacy classes of the group
\texttt{G}.\\
{\tt gap> a4:= AlternatingGroup(4);
Alt( [ 1 .. 4 ] )
gap> Order(a4);
12
gap> c:= ConjugacyClass(a4,(1,2,3));
(1,2,3)$^\wedge$G
gap> Elements(c);
[ (2,4,3), (1,2,3), (1,3,4), (1,4,2) ]
gap> cs:= ConjugacyClasses(a4);
[ ()$^\wedge$G. (1,2)(3,4)$^\wedge$G, (1,2,3)$^\wedge$G, (1,2,4)$^\wedge$G ]
gap> Elements(cs[3]);
[ (2,4,3), (1,2,3), (1,3,4), (1,4,2) ]
gap> Size(cs[3]);
4}\\
\\
Note that the notation \texttt{a$^\wedge$G} just means the set of all conjugates of \texttt{a} in
\texttt{G}. The center of a group will also be of interest; it can be found as follows.\\
{\tt gap> Center(a4);
Group(())
gap> Elements(last);
[ () ]}\\
\\
As expected the center is just the identity permutation. A nontrivial example:\\
{\tt gap> d4:= DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> Elements(Center(d4));
[ (), (1,3)(2,4) ]}\\
\\
{\bf \textsf{GAP} Project 1:}\\
\\
1. For the group $D_6$ find the elements of each conjugacy class and the size of each.\\
\\
2. Find the elements in the center of $D_6$.\\
\\
3. How do the elements in the center appear in conjugacy classes? Does this make sense?\\
\\
4. Compare the order of $D_6$ to the sum of the orders of the conjugacy classes. Make
a conjecture and explain why this should be true in general. (Note: \texttt{Order} works
with groups, subgroups, and individual elements, but \texttt{Size} must be used with other
subsets.)\\
\\
5. Verify this relationship for the groups $S_4$, $A_4\oplus Z_4$, $D_9$, $D_{10}$, and
$Z_{12}$.\\
\\
6. In the examples above, compare the size of each conjugacy class to the
order of the group. Make a conjecture about this relationship. \\
\\
{\bf \textsf{GAP} Project 2:}\\
\\
A group $G$ is a called a \emph{$p$-group} if $|G| = p^n$ for some prime $p$.
In this project, we will examine the center of a $p$-group.\\
\\
1. Of the groups we have studied, which are $p$-groups? Can you describe
all abelian $p$-groups?\\
\\
2. Use \textsf{GAP} to find the order of the center of the examples
you gave in Exercise 1. \\
\\
3. The $p$-groups we have encountered are relatively uninteresting. Other
examples are harder to build with the tools we've studied so far. However,
we will see a partial converse of Lagrange's Theorem (one of the Sylow Theorems),
that if $p^n$ divides the order of $G$, then $G$ has a subgroup $H$ of order $p^n$.
When $H$ is a $p$-subgroup of maximum order, $H$ is a called a \emph{Sylow $p$-subgroup}.
We'll learn more about these later. For now, we'll use them as a source of examples.
The command \texttt{SylowSubgroup(G,p)} will return a Sylow $p$-subgroup.\\
{\tt gap> s8 := SymmetricGroup(8);
Sym( [ 1 .. 8 ] )
gap> G := SylowSubgroup(s8,2);
Group([ (1,2), (3,4), (1,3)(2,4), (5,6), (7,8),\\ (5,7)(6,8), (1,5)(2,6)(3,7)(4,8) ])
gap> Order(G);
128
gap> Order(Center(G));
2
}\\
\noindent
4. Subgroups of $p$-groups are obviously $p$-groups. Using $G$
from the previous exercise, examine the centers of some subgroups of $G$.
You can let $H$ be any subgroup you wish. Below we pick 3 non-trivial
elements of $G$ and form the subgroup generated by them.\\
{\tt
gap> Elements(G);
gap> H := Subgroup(G, [(1,4,2,3)(5,6)(7,8), (1,7,2,8)(3,5,4,6), (1,8)(2,7)(3,6)(4,5) ]);;
gap> Order(H);
32
gap> Order(Center(H));
4
}\\
\\
5. Use \textsf{GAP} to find a Sylow $p$-subgroup $G_p$ of
$S_{10}$ for $p=2,3,5,7$. Find the order of the $G_p$ and its center.
For $p=2, 3$ also create a non-trivial proper subgroup of $G_p$
and find the order of its center.\\
\\
6. Looking at the orders of the centers of the $p$-groups you examined,
do you see a number which never occurs, even though Lagrange's Theorem
allows for this number? Make a conjecture and think about how the Class
Equation could prove your conjecture.
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