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--------------------

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 $x$, # $(G:H)$ (the ''index of $H$ in $G$;'' note that we did not discuss this in class, but it is Definition 10.13 in the text). =='"`UNIQ--h-7--QINU`"'Theorem:== # Suppose $G, H$ are groups, $\phi: G \rightarrow H$ is a homorphism. Then $\phi$ is a monomorphism iff $ker\phi = \{e_G \}$. # 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 $gkg^{-1} \in ker\phi$. # Example to show that not every subgroup can be the kernel of a homomorphism. # 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 $G$) # 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 $G$). # Theorem describing the elements of $xH$. # Theorem describing the elements of $Hx$. # Theorem relating left and right congruence when $G$ is abelian. # Example to show that the result of the previous theorem may be false when $G$ is not abelian. # Theorem relating the sizes of $xH$ and $yH$.
5. Lagrange's Theorem.