Math 407 Homework 3

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title: Homework 3
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### Directions
Solve the following problems and type up your solutions.  Your solutions should be provided in one of the following formats (in order of preference)
* typed up in $$\LaTeX$$ and submitted as a PDF on Canvas
* written legibly on blank paper, scanned into a PDF and then uploaded on Canvas
* read on video and submitted as an audition tape to a trashy reality television show

If you go with the first strategy, you may wish to check out Overleaf which is a free and intuitive website for generating $$\LaTeX$$ documents online.
If you wish to use the second method and don't own a scanner at home, you can check out the numerous scanning apps available for smartphones.

**Problem 1:**

Find all of the subgroups of $$\mathbb Z_{100}$$ and draw the associated subgroup diagram.


**Problem 2:**

Suppose that $$H\leq G$$ and $$K\leq G$$ are subgroups of $$G$$.

* (A) Show that $$H\cap K$$ must be a subgroup of $$G$$
* (B) Give an example showing that $$H\cup K$$ may not be a subgroup of $$G$$

**Problem 3:**

The rotational symmetries of a regular tetrahedron are described by the **alternating group** $$A_4$$.
It is the subgroup of $$S_4$$ defined by the *even* permutations.

$$A_4 = \{e, (12)(34), (13)(24), (14)(23), (123), (124), (132), (134), (142), (143), (234), (243)\}$$

* (A) Create a multiplication table for $$A_4$$
* (B) Show that the set $$\{(12)(34),(123)\}$$ generates $$A_4$$.
* (C) Draw the Cayley digraph of $$A_4$$ associated to this set of generators.  Try to draw it in a way to make it as neat as possible (you might have to do a couple drafts before the final version).

**Problem 4:**

Let $$S$$ be a finite set with $$n$$ elements.  A **Latin square** on $$S$$ is an $$n\times n$$ matrix $$L$$ whose entries are elements of $$S$$, where each entry of $$S$$ occurs exactly once in each row and each column.

For example, if $$S = \{1,2,3\}$$ then an example of a Latin sqsuare on $$S$$ is

$$L = \left[\begin{array}{ccc}
1 & 2 & 3\\
3 & 1 & 2\\
2 & 3 & 1
\end{array}\right]$$

* (A) Show that if $$G$$ is a group, then the multiplication table of $$G$$ defines a Latin square on $$G$$.
* (B) Conversely, suppose that $$L$$ is an $$n\times n$$ Latin square defined on a set $$A$$ of $$n$$ elements.  Viewed as a multiplication table, $$L$$ defines a binary operation on $$A$$.  What properties must our Latin square satisfy for this binary operation to make $$A$$ into a group?