Return to Homework 4, Homework Problems, Glossary, Theorems

### Problem 8

This exercise shows that there are two nonisomorphic group structures on a set of 4 elements.

Let the set be $\{e,a,b,c\}$, with $e$ the identity element for the group operation. A group table would then have to start in the manner shows in Table 4.22 of the book. The square indicated by the question mark cannot be filled in with $a$. It must be filled in either with the identity element $e$ or with an element different from both $e$ and $a$. In this latter case, it is no loss of generality to assume that this element is $b$. If this square is filled in with $e$, the table can then be completed in two ways to give a group. Find these two tables. If this square is is filled in with $b$, then the table can only be completed in one way to give a group. Find this table. Of the three tables you now have, two give isomorphic groups. Determine which two tables these are, and give the one-to-one onto renaming function which is an isomorphism.

**(a)** Are all groups of $4$ elements commutative?

**(b)** Which table gives a group isomorphic to the group $U_{4}$, so that we know the binary operation defined by the table is associative?

**(c)** Show that the group given by one of the other tables is structurally the same as the group in Exercise 14 of the book for one particular value of $n$, so that we know that the operation defined by that table is associative also.

**Solution**

Table 1:

Table 2:

(2)Table 3:

(3)Table 1 is different from 2 and 3 in that each element is its own inverse.

**(a)**

The symmetry in the main diagonal of each table shows that all groups of order $4$ are commutative.

**(b)**

Table 3 gives $U_{4}$ if $e = 1$, $a = i$, $b = -1$, and $c = -i$.

**(c)**

Let $n = 2$. Now, there are four $2 \times 2$ diagonal matrices with entries $\pm1$, which are

(4)These matrices can be used to make up Table 1.