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# Group theory: test 2

Exercises for Group theory: course.

2011:

1. Give an example of a group $G$ and a subgroup $H$ such that there is a left coset $aH$ of $H$ in $G$ not equal to the corresponding right coset $Ha.$
2. Show that a one-to-one function from a finite set to itself is also onto. What if the set is infinite?
3. Prove that $aba^{-1}b^{-1}$ is an even permutation, for any pair of permutations $a,b$.
4. Prove that the composition of two isomorphisms is an isomorphism.
5. How many automorphisms do these groups have: $\mathbf{R},\mathbf{Q},\mathbf{Z}?$
6. Prove that two cyclic groups are isomorphic if and only if they have same order.
7. Find all elements of order 3 in $\mathbf{Z}_{9}\oplus\mathbf{Z}_{12}.$
8. Find all cosets of $\mathbf{Z}$ in $\mathbf{R}.$

Older:

1. For the following permutations, determine (1) their order, (2) whether they are even or odd. $$(a)\left[ \begin{array} [{}% 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 7 & 6 & 1 & 2 & 3 & 4 & 5 \end{array} \right],$$ $$(b) (12)(134)(152),$$ $$(c)\left[ \begin{array} [{}% 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 2 & 1 & 3 & 5 & 4 & 7 & 6 & 8 \end{array} \right].$$ 2. List $D_{4}\cap A_{4}.$ Investigate this set.

3. Given a group $G,$ under what circumstances is the function $x\mapsto x^{2}$ a homomorphism, automorphism, etc.?

4. Suppose $|G|=8.$ Prove that there $G$ has an element of order 2.

5. True or false: the external direct product of two cyclic groups is cyclic.

6. Construct the Cayley table for $D_{4}/<R_{180}>.$

7. Prove that the kernel is a normal subgroup$.$