Show that the matrix is an element of order 5 in . Use this matrix to construct a nonabelian group of order 1805 and give a prepentation for this group. Classify groups of order 1805. (There are 3 isomorphism types.)

We begin with a lemma.

Lemma: Suppose and are groups and is a group homomorphism, and identify and as subgroups of . Then . Proof: Note that if , then .

We now move to the main results. Note first that .

It is straightforward to show that , , , and .

We will now construct a nonabelian group of order 1805 as a semidirect product of and . First, supposing , we write the element as the vector . Recall that ; let be the image of the matrix under the isomorphism given by . By this previous exercise, there exists a unique group homomorphism such that . Since is not trivial, is a nonabelian group of order 1805$.

Now we find a presentation for this group.

Now by the lemma, is generated by , , and . It remains to determine the pairwise products of these elements.

- Because is abelian, we have .
- Note that .
- Note that
- Note that . Thus it is straightforward to show that .

Thus this group has the presentation .

Now we classify groups of order 1805.

By FTFGAG, the abelian groups of order 1805 are (up to isomorphism) and .

Now suppose is a nonabelian group of order 1805. By Sylow’s Theorem, ; let be the unique (hence normal) Sylow 19-subgroup. Now let be a Sylow 5-subgroup of ; by Lagrange, , so that . By the recognition theorem for semidirect products, we have for some .

Now there are two groups of order up to isomorphism: and . Suppose . Now , which has no element of order 5 by Lagrange. Thus every group homomorphism is trivial, and we have . But then is abelian, a contradiction.

Thus we may assume that . Now , and this group has order . In particular, the Sylow 5-subgroups of have order 5. Now because is nontrivial and is simple, is a Sylow 5-subgroup of , and thus (by Sylow’s Theorem) is conjugate to as constructed above. Since is cyclic, by this previous exercise, .

Thus the distinct groups of order 1805 are as follows.

- , where and is any element of order 5 in .