Homework 21

Read: Section 5.1.

Please try (but do not turn in): 5.2, 5.3, 5.5, 5.6, 5.7, 5.8, 5.12, 5.14(a,b)

• Before starting, make sure to read the example at the beginning of Section 5.1 a couple of times.

• 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, so 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, which is not so obvious, 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, 5.9–5.11