Homomorphism examples

Algebra Level 3

Which of the following function(s) define group homomorphisms ?

I. $f \colon GL_2({\mathbb R}) \to GL_2({\mathbb R})$ defined by $f(A) = A^2$

II. $f \colon {\mathbb Z}_4 \to {\mathbb Z}$ defined by $f(0) = 0,$ $f(1) = 1,$ $f(2) = 2,$ $f(3) = 3$

III. $f \colon {\mathbb Z} \to {\mathbb Z}_4$ defined by $f(x) = x \pmod 4.$

Notation :

$GL_2({\mathbb R})$ is the group of invertible $2 \times 2$ matrices with real entries, with the operation being matrix multiplication.
$\mathbb Z$ is the additive group of the integers .
${\mathbb Z}_4$ is the additive group of the integers modulo 4, whose elements are $\{0,1,2,3\}.$

II and III I and II I only III only I, II, and III II only I and III None

1 solution

Patrick Corn
Jun 19, 2016

I is not a homomorphism because $f(AB) = (AB)^2 = ABAB$ and $f(A)f(B) = A^2B^2,$ and those two are not necessarily equal if $A$ and $B$ don't commute. For instance, let $A = \begin{pmatrix} 1&1\\0&1 \end{pmatrix}$ and let $B = \begin{pmatrix} 1&0 \\ 1&1 \end{pmatrix}.$ Then $ABAB = \begin{pmatrix} 5&3\\3&2\end{pmatrix}$ but $A^2B^2 = \begin{pmatrix} 5&2\\2&1\end{pmatrix}.$

II is not a homomorphism because, for instance, $f(2+2) = f(4) = f(0) = 0$ but $f(2)+f(2) = 2 +2 = 4.$

III is a homomorphism. (This is straightforward to check, and left to the reader.)

Neither the matrix A nor B that you gave as examples are invertible once their determinant is zero.

gonçalo vieira - 1 year, 5 months ago

Ah, good point. I will edit.

Patrick Corn - 1 year, 5 months ago

I have very low knowledge in Abstract Algebra. So kindly elaborate the third one

Kushal Bose - 4 years, 12 months ago

Notice that f(4×n 1 + k 1) + f(4×n 2 + k 2) = k 1 + k 2 = f(4×(n 1 + n 2) + k 1 + k 2) = f((4×n 1 + k 1) +(4×n 2 + k 2)) for each k = 0, 1, 2, or 3

Bhaskar Pandey - 1 year, 6 months ago

Homomorphism examples

1 solution

0 pending reports