189-456A: Algebra 3

Assignment 1

Due: Wednesday, September 11.

1. An anti-involution of a group $G$ is a bijection $f$ from $G$ to itself satisfying $f(ab) = f(b) f(a)$ for all $a,b\in G$.
(a) Show that every anti-involution sends the identity element of $G$ to itself.
(b) Show that the map sending $a$ to $a^{-1}$ is an anti-involution on $G$.
(c) Show that the identity is an anti-involution on $G$ if and only if $G$ is abelian.
(d) Let $G$ be a group in which every non-identity element has order $2$. Use (b) and (c) to show that $G$ is abelian. Conclude 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 $X=\{1,2,3,4\}$. Show that $S_4$ contains a subgroup of cardinality $8$. Use this to construct a homomorphism $\varphi: S_4 \rightarrow S_3$. What is the kernel of $\varphi$? Its image?

6. Let $G$ be the dihedral group of cardinality $2n$, which can be viewed as the symmetry group of the regular polygon with $n$ vertices and $n$ sides.
(a) Show that there is, up to isomorphism, a unique transitive $G$-set of size $n$ when $n$ is odd.
(b) Show that there are exactly three isomorphism classes of transitive $G$-sets of size $n$, if $n$ is even.
(c) Exhibit an explicit isomorphism of $G$ sets between the set of vertices and the set of sides of a regular $n$-gon, when $n$ is odd.
(d) Show that no such isomorphism exists, when $n$ is even.

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 extremely special and referred to as the exceptional outer automorphism of $S_6$. It is one of the small miracles which one encounters with surprising frequency in finite group theory. For more about this special feature of $S_6$, see this link