Question

Consider the set of matrices
G = a = 1 0
0 1;b = 0 1
1 0;c = 1 1
0 1;d = 1 0
1 1;e = 1 1
1 0; f = 0 1
1 1
with coefficients in Z2.
a) Make the Cayley table and check that this set forms a group with respect to matrix
multiplication. (You can assume that matrix multiplication is associative.)
b) Find the orders of all the elements in G.
c) Show that the group is isomorphic to S3 by giving an isomorphism f : G -> S3.

Answer 1

Answer on Question #44352 – Math – Abstract Algebra

Question:

Consider the set of matrices

$$G = \left\{a = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right); b = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right); c = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right); d = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right); e = \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right); f = \left( \begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array} \right); \right\}$$

with coefficients in Z2.

a) Make the Cayley table and check that this set forms a group with respect to matrix multiplication. (You can assume that matrix multiplication is associative.)

b) Find the orders of all the elements in $G$ .

c) Show that the group is isomorphic to S3 by giving an isomorphism $f: G \to S3$ .

Solution.

a) The Cayley table for $G$ is the following:

Let us see if set $G$ forms a group with respect to matrix multiplication:

- G is associative.

- neutral element or 1 is matrix a

- and every element of $G$ has inverse element, which can be easily find from the Cayley table.

b) $b^{*}b = a$ , hence the order of $b$ is 2;

$$c ^ {*} c = a, \text {hence the order of} c \text {is} 2;$$ $$d ^ {*} d = a, \text {hence the order of} d \text {is} 2;$$ $$e ^ {*} e = f; e ^ {*} e ^ {*} e = f ^ {*} e = a, \text {hence the order of} e \text {is} 3;$$

f*f=e; f*f*f=e*f=a, hence the order of f is 3.

c) The Cayley table for S3 is the following:

And comparing this two table we see that, the isomorphism between this groups is:

a -> 123

b -> 132

c -> 213

d -> 321

e -> 231

f -> 312

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