Homework 24

Read: Read Section 6.2.

• If you haven’t already, watch the following two Socratica video:
  • Normal Subgroups and Quotient Groups (aka Factor Groups) - Abstract Algebra | Socratica (11:24 min)

Turn in: 6.15, 6.22, 6.28, 6.33, 6.34(a), 6.38

• For 6.15, you’ll probably want to use 6.12 together with the general fact that $\gcd(m,n) = 1 \iff \operatorname{lcm}(m,n) = mn$. Since $|\mathbb{Z}_m \times \mathbb{Z}_n| = mn$ (by Theorem 6.6), the goal of this problem is to determine when there exists some element $(a,b) \in \mathbb{Z}_m \times \mathbb{Z}_n$, such that $|(a,b)| = mn$. Assuming $\gcd(m,n) = 1$, try to give an explicit example of $(a,b) \in \mathbb{Z}_m \times \mathbb{Z}_n$ such that $|(a,b)| = mn$; for the other direction, you want to argue that if $\gcd(m,n) \neq 1$, then $|(a,b)| < mn$ for all $(a,b) \in \mathbb{Z}_m \times \mathbb{Z}_n$.

• Before starting 6.33, read the remark below about aritmetic in quotient groups. So, for example, when you are working on part (b), you are trying to determine the smallest positive integer $k$ such that $(j\langle -1 \rangle)^k = \langle -1 \rangle$, and by the remark, this is the same as determining the smallest positive integer $k$ such that $j^k \in \langle -1 \rangle$. In part (c), you are using additive notation, so you will be looking for the smallest positive integer $k$ such that $k5 \in \langle 4 \rangle$.

• For 6.34(a), you should first use 6.31 (which follows immediately from how we stated Lagrange’s Theorem in class) to determin the order of the group $Q_8/ \langle -1 \rangle$. Knowing the order of the group will help you focus in on what familiar group it might be isomorphic to. You will probably need a little more information to decide, and computing orders of elements of $Q_8/ \langle -1 \rangle$ (like in 6.33(b)) should help.

• For 6.38, you need to work abstractly with $G/H$. To show $G/H$ is abelian, you should take arbitrary $aH,bH \in G/H$ and then prove that $(aH)(bH) = (bH)(aH)$. Remember how the operation $(aH)(bH)$ is defined.

Remark about aritmetic in quotient groups: Let $H\le G$. Remember the $G/H$ denotes the set of all left cosets of $H$ in $G$, i.e. $G/H = \{aH \mid a\in G\}$ (so $G/H$ is a set of sets).

Now, if $H\trianglelefteq G$ (i.e. if $H$ is a normal subgroup of $G$), then there is a natural binary operation that makes $G/H$ a group (and not only a set.) This is a super important idea! So, what is the binary operation on $G/H$? It is simply derived from the binary operation for $G$ as follows. For all $aH,bH\in G/H$, define

• $(aH)(bH) := (ab)H$
• $(aH)^{-1} := (a^{-1})H$
• the identity of $G/H$ is $eH = H$

So, how do we make other computations in $G/H$? For example, suppose you have a coset $aH$ and want to determin the order of $aH$ in the group $G/H$. One way to do this is to compute $aH$, $(aH)^2$, $(aH)^3$, $(aH)^4, \ldots$ until you find that $(aH)^k = H$, which is to say you keep going until $(aH)^k$ equals the identity of $G/H$. And remember that $(aH)^k$ is computed as $a^kH$. To determine when $a^kH = H$, you should look back at our notes from Section 5.2 about cosets; you should see that $a^kH = H \iff a^k\in H$.

Extra practice: 6.14, 6.16, 6.18, 6.26, 6.27, 6.29