Homework 20

Read: Section 5.1.

Turn in: 5.2, 5.3(a), 5.4(c), 5.6, 5.7, 5.8, 5.9, 5.12

• Before starting, make sure to read the example at the beginning of Section 5.1 a couple of times.
• In 5.6 and 5.7, you will need to adapt the definition of cosets to additive notation: left cosets now look like $a+H := \{a+h\mid h\in H\}$ and right cosets like $H+a := \{h+a\mid h\in H\}$.
• For 5.7, there will be only finitely many cosets, but each coset will be infinite.
• To carefully prove 5.8, you want to work with $aH$ and $Ha$ as (potentially infinite) sets. You need to show that $g\in aH$ if and only if $g\in Ha$. Also, remember that a statement like $g\in aH$ means that $g$ has a specific form: $g=ah$ for some $h\in H$.
• Note that you will not need to use any of the previous results to prove 5.12; just work through the definitions of one-to-one and onto.

Remark about working with cosets: Please carefully read over 5.9–5.11. One thing to take away is that 5.9 and 5.10 together yield an important criteria for determining when two left (respectively right) cosets are equal:

\[aH = bH \iff b^{-1}a \in H \iff a^{-1}b\in H.\] \[Ha = Hb \iff ab^{-1}\in H \iff ba^{-1} \in H.\]

You are welcome to use this!

Extra practice: 5.4(a),5.5, 5.10–5.11

• To help you check your work, here’s part of the computation for 5.4(a). The group is $Q_8$ one of the subgroups is \(H = \langle i \rangle = \{1,-1,i,-i\}\). To compute the left cosets of $H$ in $Q_8$, we take individual elements of \(Q_8\) and multiply $K$ on the left by them. (Remember: $ij = k$, $ji = -k$, $ik = -j$, $ki = j$, $jk=i$, $kj = -i$.)
  • \(1H = \{1\cdot1,1\cdot(-1), 1\cdot i, i\cdot (-i)\} = \{1,-1, i, -i\} = H\)
  • \(-1H =\) (you do it)
  • \(iH = \{i\cdot1,i\cdot(-1), i\cdot i, i\cdot (-i)\} = \{i,-i, -1, 1\} = H\)
  • \(-iH =\) (you do it)
  • \(jH = \{j\cdot1,j\cdot(-1), j\cdot i, j\cdot (-i)\} = \{j,-j, -k, k\}\)
  • \(-jH =\)(you do it)
  • \(kH =\)(you do it)
  • \(-kH = \{(-k)\cdot1,(-k)\cdot(-1), (-k)\cdot i, (-k)\cdot (-i)\} = \{-k,k,-j, j\} = jH\)

  In the end, for this part, you should only have two different cosets of $H$ in $Q_8$ (the rest are repeats).