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# Group theory: final exam

Exercises for Group theory: course.

2011:

1. From the definition of a group, prove that the identity and the inverse of an element is unique.
2. Provide a full description of (a) the symmetry group of the square, (b) the group of rotations of the circle.
3. (a) Define cosets. (b) Prove that two distinct left cosets are disjoint. (c) Give an example of two cosets that aren't disjoint.
4. (a) List the cosets in $\mathbf{Z}/<3>.$ (b) Provide the Cayley table for this group.
5. Give a group-theoretic interpretation of the relation between the derivative and anti-derivative.
6. Prove that an homomorphism is one-to-one if and only if its kernel is trivial.
7. (a) State the First Isomorphism Theorem. (b) Prove that the isomorphism is well defined. (c) Give an example of how the theorem is applied.
8. List (a) all abelian groups of order 54; (b) all abelian groups of order 100; (c) all groups of order 4.
9. (a) Define the external direct product of $n$ groups. (b) State and prove the theorem about the orders of elements in the product of two groups.

Old:

1. Give examples of the following. The example must be different from all groups in this exam. (a) A group of numbers with respect to multiplication, (b) a group of $2\times2$ matrices with respect to multiplication, (c) a cyclic group of symmetries other then rotations.

2. From the definition of a group, prove that the identity and the inverse of an element is unique.

3. Which of these groups are cyclic? For each cyclic group list all generators of the group. (a) $\mathbf{Q};$ (b) $6\mathbf{Z};$ (c) $\{6^{n}:n\in\mathbf{Z}\}$ under multiplication$;$ (d) $\{a+b\sqrt{2} :a,b\in\mathbf{Z}\}$.

4. Find the maximal possible order of an element in (a) $S_{5},$ (b) $S_{10}.$

5. Suppose $H$ is a subgroup of $G.$ (a) Prove that all left cosets of $H$ in $G$ have the same size. (b) Derive the Lagrange theorem.

6. Find all values of $a$ such that the cyclic subgroup in $M_{2}% (\mathbf{R})$ generated by $A=\left[ \begin{array} [{}% 0 & a\\ -a & 0 \end{array} \right]$ is finite.

7. (a) State the Fundamental Theorem of Finite Abelian Groups. (b) Find all abelian groups, up to isomorphism, of order 500.

8. Find all subgroups of $\mathbf{Z}_{2}\oplus\mathbf{Z}_{3}\oplus \mathbf{Z}_{6}.$

9. For each of the following sets with two binary operations, decide whether it forms a ring. (a) $2\mathbf{Z};$ (b) $\mathbf{Z}^{+}$; (c) $2\mathbf{Z}\oplus\mathbf{Z}$; (d) $\{a+b\sqrt{2}:a,b\in\mathbf{Z\}.}$ If it is a ring, state whether it has a unity, is commutative, there are elements that are not units.

10. Show that a group that has only finite number of subgroups must be finite.