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Abstract Algebra - Graduate Course

Groups and Subgroups

For discussion:

Products

  1. If $G$ and $H$ are groups, then $G\times H$ is a group where $(a,b)(c,d)=(ac,bd)$.
  2. The product of abelian groups is abelian.
  3. If $G$ and $H$ are groups, then $G\times H$ is isomorphic to $H\times G$.
  4. Let $G=\mathbb{Z}/2\mathbb{Z}$ and $H=\mathbb{Z}/3\mathbb{Z}$. Prove that $G\times G$ has $4$ elements but is not isomorphic to $\mathbb{Z}/4\mathbb{Z}$, and $G\times H$ has $6$ elements and is isomorphic to $\mathbb{Z}/6\mathbb{Z}$.

Subgroups

  1. The intersection of (arbitrarily many) subgroups of $G$ is a subgroup of $G$.
  2. If $G$ is abelian and $n>1$ is an integer, then the set $G(n)=\{x^{n}: x\in G\}$ is a subgroup of $G$. What if $G$ is not abelian?
  3. If $A$ is a subgroup of $G$ and $B$ is a subgroup of $H$ then $A\times B$ is a subgroup of $G\times H$.
  4. $G$ is isomorphic to the set of diagonal elements $(g,g)$ in $G\times G$.
  5. Using the notation in (2), let $G=\mathbb{Z}/10\mathbb{Z}$. What is $G(3)$? What is $G(2)$?

Dihedral and Symmetric Groups

  1. If there is a bijection from $X$ to $Y$, prove that $S(X)$ is isomorphic to $S(Y)$.
  2. Review the rules for cycle decomposition and multiplication of permutations in $S_n$. Let $\sigma=(4\ 8\ 6)(4\ 3\ 8)(1\ 3\ 5\ 2)$.
    • Write $\sigma$ as a product of disjoint cycles.
    • Write $\sigma$ as a product of transpositions. What is its sign?
  3. The group of symmetries of the square $D_{8}$ “comes with” an action on the $4$ corners of the square. Labelling the four corners with $X=\{1,2,3,4\}$, this action associates a permutation of $X$ to each element of $D_{8}$. Show that this is an injective homomorphism from $D_{8}$ to $S_{4}$ and compute it for each element of $D_{8}$. What can you say about the general case?

Matrix Groups

  1. Let $G=\GL_{2}(\mathbb{R})$. Describe some proper subgroups of $G$. Include some finite and some infinite examples.
  2. Let $G=\GL_{2}(\mathbb{Z}/2\mathbb{Z})$ be the set of two by two matrices with entries in $\mathbb{Z}/2\mathbb{Z}$. Verify that $G$ is a group of order $6$. It acts on the set $\Zn{2}\times\Zn{2}$ by matrix multiplication. Show that it preserves the set of $3$ vectors $(a,b)$ where not both $a$ and $b$ are zero, and that $G$ is isomorphic to $S_{3}$.