Math 361, Spring 2022, Assignment 1

Read:

1. Section 14.

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

1. $H\leq G$
2. $H\trianglelefteq G$.
3. Coset multiplication (when $H\trianglelefteq G$).
4. Canonical projection (from $G$ onto $G/H$).

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

1. Theorem characterizing when coset multiplication is well-defined.
2. Theorem concerning the properties of coset multiplication ("When $H\trianglelefteq G$, coset multiplication turns $G/H$ into a...").
3. Theorem describing the kernel of the canonical projection.
4. Fundamental Theorem of Homomorphisms.

Solve the following problems:

1. Make an operation table for the quotient group $\mathbb{Z}_{12}/\left\langle 4\right\rangle$.
2. Section 14, problems 1, 9, 24, 30 (hint: in December we used Lagrange's Theorem to prove that for any finite group $G$ and any $g\in G$, one always has $g^{\left\lvert G\right\rvert}=e$; now apply similar reasoning to the group $G/H$), and 31.
3. Consider the group $D_4$ of symmetries of a square; for purposes of this problem we will use the notation introduced on page 80 of the text. Let $H$ denote the subgroup $\left\langle\delta_1,\delta_2\right\rangle$ generated by reflections in the diagonals. Determine whether $H$ is a normal subgroup of $D_4$. If it is a normal subgroup, then write the operation table for the quotient group $D_4/H$. (Warning: normality can be checked by brute force, but this is very tedious and there is a shorter way. In the next problem you will actually need the brute-force check but it will be much shorter.)
4. Repeat the above exercise for the subgroup $\left\langle\delta_1\right\rangle$ generated by $\delta_1$ alone.
5. Let $\pi:\mathbb{Z}_{12}\rightarrow\mathbb{Z}_{12}/\left\langle 4\right\rangle$ denote the canonical projection. Write the table of values for $\pi$.
6. Let $G$ be any group. Prove that the trivial subgroup $\{e\}$ is normal in $G$. Then prove that the quotient group $G/\{e\}$ is isomorphic to $G$ itself. (Hint: you need to show that the canonical projection is injective in this case.)
7. Let $G$ be any group. Prove that the improper subgroup $G$ is normal in $G$. Then write the operation table for the quotient group $G/G$.
8. Recall that $\mathbb{R}^*$ denotes the set of non-zero real numbers, regarded as a group under ordinary multiplication, and let $\phi:S_n\rightarrow\mathbb{R}^*$ denote the sign homomorphism that takes even permutations to $1$ and odd permutations to $-1$. Compute $\ker(\phi)$, write an operation table for the quotient group $S_n/\ker(\phi)$, and give a table of values for the momomorphism $\widehat{\phi}:S_n/\ker(\phi)\rightarrow\mathbb{R}^*$ whose existence is asserted by the Fundamental Theorem.

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Solutions:

Complete Notes:

https://drive.google.com/file/d/1wCXbcBMd8br--G0xReIPzf8FN9sTtfHU/view?usp=sharing

Definitions:

1. $H\leq G$: $H$ is a subgroup of $G$.
2. $H\trianglelefteq G$: H is a normal subgroup of G. (When $\sim_{l,H}$ and $\sim_{r,H}$ are the same relation: if $\forall g \in G, \forall h \in H, ghg^{-1} \in H$
3. Coset multiplication (when $H\trianglelefteq G):(g_{1}H)(g_{2}H) = (g_{1}g_{2})H$.
4. Canonical projection (from $G$ onto $G/H$): Suppose $H\trianglelefteq G$. Define $\pi : G \rightarrow G/H$ as follows: $\pi (a) = aH$. Then $\pi$ is an epimorphism.

Theorems:

1. Theorem characterizing when coset multiplication is well-defined: If H is a normal subgroup of G, Then coset multiplication is well-defined.
2. Theorem concerning the properties of coset multiplication ("When $H\trianglelefteq G$, coset multiplication turns $G/H$ into a..."): If $H\trianglelefteq G$, then $G/H$ is a group under coset multiplication.
3. Theorem describing the kernel of the canonical projection: Let $\pi : G \rightarrow G/H$ be the canonical projection. Then $ker(\pi) = H$.
4. Fundamental Theorem of Homomorphisms: Suppose $\phi : G \rightarrow H$ is a homomorphism. Let $\phi : G \rightarrow G/Ker(\phi)$ be the canonical projection. Then there exists a unique monomorphism $\widehat{\phi}: G/Ker(\phi) \rightarrow H$ such that $\widehat{\phi} \circ \pi = \phi$

Problems:

1. $0+ \left\langle 4 \right\rangle$ = $\{0, 4, 8\}; 1+ \left\langle 4 \right\rangle$ = $\{1, 5, 9\} \cdots$. operation table: A table of $\mathbb Z_4$.

Book Problem 1. $\mathbb Z_6 /\left\langle 3 \right\rangle$ $\left\langle 3 \right\rangle = \{0, 3\}$. Order $= \frac{6}{2} = 3$

Book Problem 9. 4

Book Problem 24. 2, $\mathbb Z_2$

Book Problem 30. $|G/H| = m, (aH)^{m} = eH = H, (aH)^{m} = (a^m)H, a^m \in H$.

Book Problem 31. $k \in K$, $K$ is the intersection of all normal subgroups. $gkg^{-1} \in$ all normal subgroups, therefore, $gkg^{-1} \in$ intersection of all normal subgroups, $gkg^{-1} \in K$. K is a normal subgroup.

3. Yes, 4

4. yes, 2

5. $\pi (0) = 0 + \left\langle 4 \right\rangle, \pi (1) = 1 + \left\langle 4 \right\rangle, \pi (2) = 2 + \left\langle 4 \right\rangle, \pi (3) = 3 + \left\langle 4 \right\rangle$

6. $aE = {a}, bE = {b}, aE = bE, {a}={b}, a == b$

7. $\forall g_1 \in G, g_2 \in G, g_1 g_2 g_1^{-1} \in G$ because of closure. $G/G: \{G\}$