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# Group theory: final exam

Exercises for Group theory: course.

2011:

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