189-456A: Algebra 3

Assignment 1

Due: Wednesday, September 13.

1. Let $G$ be a group in which every non-identity element has order $2$. Show that $G$ is abelian. Use this to show that every group of cardinality $4$ is abelian, and classify all such groups up to isomorphism.

2. Let $G$ be a group of cardinality $6$. Show that there is a transitive $G$-set of cardinality $3$. Use this to classify all the possible groups of cardinality $6$, up to isomorphism.

3. Use problems $1$ and $2$ to give a complete list of all the groups of cardinality $\le 7$, up to isomorphism.

4. Let $G_1= \{ 1,-1,i,-i,j,-j,k,-k\}$ be the subgroup of the multiplicative group of Hamilton's quaternions, satisfying the relations $$ i^2 = j^2 = k^2=-1, \quad ij =-ji = k, \quad ki = -ik = j, \quad jk=-kj=i,$$ and let $G_2$ be the dihedral group of order $8$. Are $G_1$ and $G_2$ isomorphic? Explain.

5. Let $S_4$ be the permutation group on the set $X=\{1,2,3,4\}$. A $(2,2)$-partition of $X$ is an unordered pair $\{ A, B\}$ of subsets of $X$ of cardinality $2$ for which $X = A \cup B$. Show that the set $\Sigma$ of $(2,2)$-partitions has cardinality $3$, and use this to construct a homomorphism $\varphi: S_4 \rightarrow S_3$. What is the kernel of $\varphi$? Its image?

6. Let $X$ be a square in the plane and let $V$ and $E$ be the set of vertices and edges of $X$, equipped with their natural action of $D_8 = {\rm Aut}(X)$.
(a) Show that $V$ and $E$ are not isomorphic as $D_8$-sets.
(b) Show that any transitive $D_8$-set of cardinality $4$ is isomorphic either to $V$ or to $E$.
As several of you pointed out, there is a mistake in (b) and there is in fact a third transitive $D_8$-set which is not isomorphic to those other two. It is somewhat degenerate because the action of $D_8$ on it is not faithful. (An action of a group $G$ on a set $X$ is said to be faithful if the resulting homomorphism from $G$ to the group of permutations on $X$ is injective. The faithful actions are the ones that allow you to realise $G$ as a subgroup of the permutation group on $X$.)

7. Show that $S_5$ contains a subgroup of cardinality $20$. (Hint: consider the set of affine linear transformations $x \mapsto ax+b$ with $a\in \mathbb F_5^\times$ and $b\in \mathbb F_5 = \mathbb Z/5\mathbb Z$ belonging to the field with $5$ elements.)

8. Use the results of question $7$ to construct a set $X$ of cardinality $6$ on which the group $S_5$ acts transitively.

9. Using question 8, show that $S_6$ contains at least two distinct conjugacy classes of subgroups that are isomorphic to $S_5$.

10. Use the result of question 9 to construct an automorphism of $S_6$ which is not an inner automorphism, i.e., show that the natural map $S_6 \rightarrow {\rm Aut}(S_6)$ given by the conjugation action of $S_6$ on itself is not surjective (although it is injective.)

Remark: For all $n>6$ the permutation group $S_n$ is isomorphic to its automorphism group: the homomorphism $S_n\rightarrow {\rm Aut}(S_n)$ is injective (since $S_n$ is simple!) and every automorphism of $S_n$ is in fact an inner automorphism. The automorphism constructed in Q. 10 is therefore special and referred to as the exceptional outer automorphism of $S_6$.