Math 360, Fall 2018, Assignment 15

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

Carefully define the following terms, then give one example and one non-example of each:

1. Kernel (of a homomorphism).

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning images and preimages of subgroups under homomorphisms.
2. Theorem relating injectivity to kernels.
3. Theorem concerning normality of kernels.
4. Fundamental theorem of homomorphisms.

Solve the following problems:

1. Recall the sign mapping $\mathrm{sgn}:S_n\rightarrow\mathbb{R}^*$ that sends even permutations to $1$ and odd permutations to $-1$. Although we did not use the language of homomorphisms at the time, we showed in class that $\mathrm{sgn}$ is a homomorphism from $S_n$ to $(\mathbb{R}^*,\cdot)$.

(a) Describe the kernel of $\mathrm{sgn}$.

(b) Make a group table for the quotient $S_n/\ker(\mathrm{sgn})$.

(c) Make a group table for the image $\mathrm{im}(\mathrm{sgn})$.

(d) Explicitly calculate the isomorphism $\widehat{\mathrm{sgn}}:S_n/\ker(\mathrm{sgn})\rightarrow\mathrm{im}(\mathrm{sgn})$ described by the fundamental theorem of homomorphisms.

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