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Week 13 problems

These problems are taken (mostly) from the Chapter 15 Exercises in the text.

  1. (Problem 3) Show that every group of order $45$ has a normal subgroup of order $9$.

  2. (Problem 6) Prove that a group of order $160$ has a proper normal subgroup and is therefore not simple.

  3. (Problem 12) Let $G$ be a group of order $p^r$ where $p$ is a prime. Prove that $G$ has a normal subgroup of order $p^{r-1}$.

  4. (Problem 20) What is the smallest odd number $n$ for which there is a nonabelian group $G$ with $n$ elements? Can you find such a group?

  5. (Problem 18 simplified) Let $G$ have order $p^aq^b$ where $p$ and $q$ are primes. If $G$ has only one Sylow $p$-subgroup $P$ and one Sylow $q$-subgroup $Q$ then $G$ is isomorphic to $P\times Q$.

  6. Let $G$ be a group with $2m$ elements where $m$ is odd. We will prove that $G$ has a non-trivial normal subgroup of index $2$.

    • By Cayley’s theorem, $G$ acts as a set of permutations of itself by the action $(g,g’)\mapsto gg’$.

    • Show that $G$ contains an element $\sigma$ of order $2$.

    • Show that $\sigma$ acts on $G$ as a product of $m$ transpositions.

    • Show that the homomorphism $f:G\to \mathbb{Z}_{2}$ obtained by taking the sign of the permutation of $g$ acting on itself is surjective.

    • Conclude that $G$ has a normal subgroup of index $2$.