Algebra 1. Seminar 2

Problem 1. a) Let $A$ and $B$ be abelian groups. Consider the set of homomorphisms $\mathrm{Hom}(A,B)$ and define on it the operation $(\alpha +\beta )(x)=\alpha (x)+\beta (x)$. Prove that the set of homomorphisms is a group under this operation. b) Find the group $\mathrm{Hom}(\mathbb{Z}/m\mathbb{Z},\mathbb{Z}/n\mathbb{Z})$ for $m,n\ge 0$.

Problem 2. Let $F\subset H\subset G$ be a chain of subgroups (not necessarily normal). Show that $(G:F)=(G:H)(H:F)$.

Problem 3. a) Now let $F\subset H\subset G$ with $F$ and $H$ normal in $G$. Prove that $G/H\simeq (G/F)/(H/F)$ (in particular, all the quotients make sense). b) Is it enough to require only that $F$ be normal in $H$?

Problem 4. a) Find a group $G$ and two isomorphic subgroups $G_1$ and $G_2$ of it such that $G/G_1\not\simeq G/G_2$. b) Find two non-isomorphic groups that have isomorphic subgroups with isomorphic quotients.

Problem 5. Prove that two permutations in $S_n$ are conjugate if and only if they have the same cycle type (that is, the same cycle lengths in the decomposition into disjoint cycles).

Problem 6. Find the conjugacy classes of the group $A_5$ (it is enough to give one representative of each class).

Problem 7. Prove that for $n>2$, $n\ne 6$ there is an isomorphism $\mathrm{Aut}(S_n)\simeq S_n$, and all automorphisms are inner (hint: an automorphism takes conjugacy classes to conjugacy classes. Where can the class of transpositions go?).

Problem 8. Prove that the rotation group of the cube is isomorphic to $S_4$ (hint: consider the action on certain elements of the cube), and that the symmetry group of the cube is isomorphic to $S_4\times \mathbb{Z}/2\mathbb{Z}$.

Problem 9. Construct a surjective homomorphism $S_4\to S_3$ and find its kernel.

Problem 10. Find the order of the symmetry group of the icosahedron.

Problem 11. Find the order of the automorphism group of the graph

Problem 12. Show that this graph can be constructed as follows: the vertices are the three-element subsets of $\{1,2,3,4,5\}$, and two of them are joined by an edge if and only if they have exactly one element in common. Using this, find the symmetry group of the graph.