AUMAT 229 Activities

(a)

$\phi :\mathbb{Z}\to {\mathbb{Z}}_n$ defined by taking $\phi (k)$ to be the residue of $k$ modulo $n\text{.}$

(b)

$\phi :{\mathbb{Z}}_3\to {\mathbb{Z}}_6$ defined by setting

(c)

$\phi :{\mathbb{Z}}_6\to D_6$ defined by $\phi (k)=r^k\text{.}$

(d)

$det:{\mathrm{GL}}_n(\mathbb{R})\to {\mathbb{R}}^{\times }\text{.}$

(e)

$\mathrm{sgn}:S_n\to \{\pm 1\}\text{,}$ where $\mathrm{sgn}(p)$ is the sign of the permutation $p\text{,}$ either $1$ for an even permutation or $-1$ for an odd permutation.

(Note: the group $\{\pm 1\}$ is the multiplicative version of the cyclic group ${\mathbb{Z}}_2\text{.}$)

(f)

$\mathrm{trace}:{\mathrm{M}}_n(\mathbb{R})\to \mathbb{R}\text{,}$ where ${\mathrm{M}}_n(\mathbb{R})$ is the additive group of $n\times n$ matrices (both invertible and singular), and the trace of a square matrix is defined to be the sum of its diagonal entries.

(g)

$\iota :\mathbb{Z}\to \mathbb{Q}$ defined by $\iota (m)=\frac{m}{1}\text{.}$

(h)

$\iota :{\mathbb{Q}}^{\times }\to {\mathbb{R}}^{\times }$ defined by taking $\iota (a/b)$ to be the decimal number that results from performing long division of $b$ into $a\text{.}$

(Note: You may skip the operation-preserving check for this example.)

(i)

$\phi :{\mathbb{R}}^2\to \mathbb{R}$ defined by taking $\phi (\mathbf{v})$ to be the (oriented) distance from the origin to the orthogonal projection of $\mathbf{v}$ onto the line $y=x\text{,}$ where positive distances are to the upper right and negative distances are to the lower left.

(Note: You should perform the check that this map is operation-preserving geometrically. You still remember how to draw a geometric diagram of vector addition in ${\mathbb{R}}^2\text{,}$ right?)

(j)

$H$ is a normal subgroup of group $G\text{,}$ and $\rho :G\to G/H$ is the “projection” map defined by

That is, $\rho$ sends each element in $G$ to the coset in $G/H$ that contains it.

(a)

Verify that

(b)

Now multiply both sides of (✶) by ${(\phi (e))}^{-1}$ and simplify to arrive at the desired conclusion.

Warning: Do not change ${(\phi (e))}^{-1}$ into $\phi (e^{-1})\text{,}$ as we have not yet verified that homomorphisms preserve inverses.

(a)

Unfortunately the reasoning of Discovery 7.7 does not hold for homomorphisms. Use one of the example homomorphisms in Discovery 16.1 to demonstrate a specific example of an element $x$ in $G$ so that $x$ and $\phi (x)$ have different orders.

(b)

Use Discovery 7.6 to verify that if $x$ has order $n\text{,}$ then

will hold. So what can be said about the relationship between the order of $x$ and the order of $\phi (x)\text{?}$

(a)

Verify both of the equalities

Warnings: Do not assume that $G^{\prime }$ is Abelian, so the second equality does not follow from the first. Also, do not change $\phi (x^{-1})$ into ${(\phi (x))}^{-1}\text{,}$ as that would be assuming our desired final conclusion.

(b)

Explain why the equalities established in Task a imply that we must have

(a)

Closed under multiplication. Suppose $y_1,y_2$ are elements of $\mathrm{im}\phi \text{.}$ By definition, this means that there are corresponding elements $x_1,x_2$ in $G$ so that

Your task is to now demonstrate that the product $y_1{y}_2$ is also in $\mathrm{im}\phi$ by filling in the blank in the equality

with some element of $G\text{.}$

(b)

Closed under taking inverses. Suppose $y$ is an element of $\mathrm{im}\phi \text{.}$ By definition, this means that there is a corresponding element $x$ in $G$ so that

Your task is to now demonstrate that the inverse $y^{-1}$ is also in $\mathrm{im}\phi$ by filling in the blank in the equality

with some element of $G\text{.}$

(a)

Prove that if $\phi$ is actually an isomorphism, then $\ker \phi$ contains only $e\text{.}$

(b)

Prove the converse: If $\ker \phi$ contains only $e\text{,}$ then $\phi$ must be an isomorphism.