Algebra 1. Seminar 1

Problem 1. By filling in the multiplication table, classify all groups of order $3$ and $4$.

Problem 2. a) Prove that if every element of a group has order $2$, then the group is abelian. b) Suppose this group is finite. Without using the structure theorem for abelian groups, prove that it is isomorphic to a direct sum (or direct product) of copies of $\mathbb{Z}/2\mathbb{Z}$.

Problem 3. Classify all groups of order $6$ and $10$.

Problem 4. a) Give a presentation of the group $D_n$ by generators and relations. b) A group is given by generators and relations, $G=⟨a,b\mid a^2=b^2=(ab{)}^2,a^4=1⟩$. How many elements does it have?

Problem 5. a) Show that the groups $\mathbb{Q}$ and ${\mathbb{Q}}^{\ast }$ are not finitely generated. b) Prove that the group $\mathbb{Q}$ cannot be written as a nontrivial direct product. c) Write the group ${\mathbb{Q}}^{\ast }$ as an infinite direct sum of cyclic groups.

Problem 6. For the group $D_n$ find a) the centre (that is, the set of elements that commute with all elements), b) all subgroups, c) all normal subgroups and the quotients by them.

Problem 7. a) Prove that the group $D_4$ is not a direct sum of its nontrivial subgroups. b) Give an example of a group $D_n$ that is a direct sum of its nontrivial subgroups.

Problem 8. Let $m$ and $n$ be coprime natural numbers. Prove that there is an isomorphism $\mathbb{Z}/mn\mathbb{Z}\simeq \mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$. Show that this statement is never true if $m$ and $n$ are not coprime.

Problem 9. a) How many distinct cycles of length $k$ are there in $S_n$? b) Find all cycle types (types of decomposition into disjoint cycles) in the group $S_6$. c) For each cycle type, find how many permutations in $S_6$ have that type. d) Find all numbers that occur as the order of some element of $S_{10}$.

Problem 10. Prove that every permutation can be written as a product of a) elementary transpositions $(i,i+1)$, $1\le i\le n-1$; b) $(1,n)$ and $(1,2,3,\dots ,n)$.

Problem 11. Prove that the group of inner automorphisms of a group is normal in the group of all its automorphisms.

Problem 12. Compute the automorphism groups of a) $\mathbb{Z}$, b) $\mathbb{Z}/n\mathbb{Z}$, c) $S_3$, d) $S_4$.