Math 360, Fall 2021, Assignment 14

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

Read:

1. Section 10.

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

1. $\sim_{l,H}$ (the relation of left congruence modulo the subgroup $H$).
2. $\sim_{r,H}$ (the relation of right congruence modulo the subgroup $H$).
3. $xH$ (the left coset of $H$ by $x$).
4. $Hx$ (the right coset of $H$ by $x$).
5. $(G:H)$ (the index of $H$ in $G$; note that we did not discuss this in class, but it is Definition 10.13 in the text).

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

1. Theorem concerning a special property of kernels, not shared by arbitrary subgroups.
2. Example to show that not every subgroup can be the kernel of a homomorphism.
3. Theorem concerning the properties of the left congruence relation ("$\sim_{l,H}$ is an...").
4. Theorem concerning the properties of the right congruence relation ("$\sim_{r,H}$ is an...").
5. Theorem describing the elements of $xH$.
6. Theorem describing the elements of $Hx$.
7. Theorem relating left and right congruence when $G$ is abelian.
8. Example to show that the result of the previous theorem may be false when $G$ is not abelian.
9. Theorem relating the sizes of $xH$ and $yH$.
10. Lagrange's Theorem.

Solve the following problems:

1. Section 10, problems 1, 3, 6, 7, 9, 10, 12, 15, 20, 21, 22, 23, and 24.

--------------------End of assignment--------------------

Questions:

Theorem relating the sizes of $xH$ and $yH$.

1. Let $μ = (1,2,4,5)(3,6)$ in $S_6$. Find the index of $⟨μ⟩$ in $S_6$.

2.A subgroup of a group of order 6 whose left cosets give a partition of the group into 12 cells. Cannot be done, index = 12, 6/12=1/2

3. Example to show that not every subgroup can be the kernel of a homomorphism.

Definitions:

1. $\sim_{l,H}$ (the relation of left congruence modulo the subgroup $H$): Let $G$ be any group, $H$ be any subgroup of $G$. Define a binary relation $\sim_{l, H}$ (left congruence modulo $H$) by $x \sim_{l, H} y \iff y^{-1}x \in H$.
2. $\sim_{r,H}$ (the relation of right congruence modulo the subgroup $H$):Define a binary relation $\sim_{r, H}$ (right congruence modulo $H$) by $x \sim_{r, H} y \iff xy^{-1} \in H$.
3. $xH$ (the left coset of $H$ by $x$): $[x]_{l,H}$ is abbreviated as $xH$, and is called the left coset of $H$ by x, but often just the left coset of $x$. ($xH = \{ xh|h \in H \}$). Example: $G = S_3, H = <(12)> = \{\iota, (1,2)\}$. All left cosets: $H = \{\iota, (12)\}, \iota H = \iota \{\iota, (12)\} = \{\iota, (12)\}$ $= (12)H = (12)\{\iota, (12)\} = \{(1,2), \iota \}$, $(13)\{\iota, (12)\} = \{(13),(123)\}$, $(23)\{\iota, (12)\} = \{(23),(132)\}$, $(123)\{\iota, (12)\} = \{(123),(13)\}$.
4. $Hx$ (the right coset of $H$ by $x$): $[x ]_{r,H}$is abbreviated as $Hx$, and is called the right coset of $H$ by $x$, but often just the right coset of $x$. ($Hx = \{ hx|h \in H \}$).
5. $(G:H)$ (the index of $H$ in $G$;):Let $H$ be a subgroup of a group $G$. The number of left cosets of $H$ in $G$ is the index $(G : H)$ of $H$ in $G$. If $G$ is finite, then obviously $(G : H)$ is finite and $(G : H) = |G|/|H|,$ since every coset of $H$ contains $|H|$ elements.

Theorem:

1. Suppose $G, H$ are groups, $\phi: G \rightarrow H$ is a homorphism. Then $\phi$ is a monomorphism iff $ker\phi = \{e_G \}$.
2. Theorem concerning a special property of kernels, not shared by arbitrary subgroups: $\phi: G \rightarrow H$ is a homorphism, and $k \in ker\phi$. let $g \in G$ be arbitrary. Then $gkg^{-1} \in ker\phi$.
3. Example to show that not every subgroup can be the kernel of a homomorphism: $G=S_3, K=<(1 2)>$. Take $k=(1 2), g = (1 2 3)$, $<(1 2)> = \{\iota, (1 2)\}$, $gkg^{-1} = (1 2 3)(1 2)(1 2 3)^{-1} = (1 2 3)(1 2)(1 3 2) = (1)(2 3)=(3 2)$, so $gkg^{-1} \not\in K$.
4. Theorem concerning the properties of the left congruence relation ("$\sim_{l,H}$ is an..."): $\sim_{l,H}$ is an equivalence relation on $G$. (and defines a partition of $G$)
5. Theorem concerning the properties of the right congruence relation ("$\sim_{r,H}$ is an...").$\sim_{r,H}$ is an equivalence relation on $G$. (and defines a partition of $G$).
6. Theorem describing the elements of $xH$: $xH = \{xh | h\in H\}$
7. Theorem describing the elements of $Hx$: $Hx = \{hx | h \in H\}$
8. Theorem relating left and right congruence when $G$ is abelian: $xH = Hx$ $\sim_{l,H}, \sim_{r,H}$ are the same. Also, $x+H, H+x$ are the same.
9. Example to show that the result of the previous theorem may be false when $G$ is not abelian: $a+2b != b + 2a$
10. Theorem relating the sizes of $xH$ and $yH$: They are equinumerous.
11. Lagrange's Theorem: Let $G$ be a finite group, and suppose $H <=G$. Then $|H|$ is a divisor of $|G|$.

Book Problems:

1, 3, 6, 7, 9, 10, 12, 15, 20, 21, 22, 23, 24

1. $4\mathbb Z = \{\cdots, -8, -4, 0, 4, 8, \cdots\}, 1+4\mathbb Z = \{\cdots , -7, -3, 1, 5, 9, \cdots \}$, $\{2+4\mathbb Z = \{\cdots, -6, -2, 2, 6, 10 \cdots\}$, $3+4\mathbb Z = \{\cdots, -5, -1, 3, 7, 11, \cdots \}$

3. $<2> = \{0, 2, 4, 6, 8, 10\}, 1 + <2> = \{1,3,5,7,9,11\}$

6.$\{\rho_0, \mu_2\}, \{\rho_1, \delta_2\}, \{\rho_2, \mu_1\}, \{\rho_3, \delta_1\}