Group theory: test 1

Grading

Tests for Group theory: course.

2011:

1. Let $x\in\mathbf{R}$ with $x>-1$ and $x\neq0.$ Show that the inequality $(1+x)^{n}>1+nx$ holds for every positive integer $n\geq2.$
2. Let $X=\{n^{2}:n\in\mathbf{Z},n\geq0\}.$ Determine whether $X$ is closed under (a) addition and (b) multiplication.
3. Show that every group $G$ with property $xx=e$ for all $x\in G$ is abelian. Hint: consider $(ab)(ab).$
4. Describe the group of symmetries of (a) the hyperbola $y=1/x,$ (b) the parabola $y=x^{2}$.
5. Give the multiplication table for the set $\{e,a,b\}$ of three elements satisfying axioms 2 and 3 but not 1 (associativity).
6. Determine whether the following sets of invertible matrices is a subgroup of $GL(n,\mathbf{R}):$ (a) matrices with determinant $2;$ (b) diagonal matrices; (c) matrices with only 0s on the diagonal; (d) matrices with determinant equal to $1$ or $-1$. Just yes or no.
7. (a) Find the order of the cyclic subgroup of $\mathbf{Z}_{30}$ generated by $25$. (b) Find the order of the cyclic subgroup of $\mathbf{Z}_{42}$ generated by $30$.
8. Find the subgroup lattice for $\mathbf{Z}_{18}.$
9. Prove that in a finite group, $|ab|=|ba|.$

Old:

1. For every positive integer $n,$ prove that a set with exactly $n$ elements has exactly $2^{n}$ subsets (counting the empty set and the set itself).
2. Construct the Cayley table of the groups of symmetries of (a) non-circular ellipse, and (b) Crysler's logo.
3. Using only the definition prove that $(\mathbf{Z}_{n},+)$ is a group.
4. Prove the right cancellation law.
5. Prove that the centralizer $C(a)$ of an element $a$ of a group $G$ is a subgroup of $G$.
6. Determine the subgroup lattice for $\mathbf{Z}_{12}.$
7. Find all generators of $\mathbf{Z}.$