Tour:Mind's eye test five (beginners)

Symmetric groups on sets of small size

 * 1) Prove that the symmetric group of degree two is isomorphic to the cyclic group of order two.
 * 2) Prove that the symmetric group of degree $$n$$, for $$n \ge 3$$, is non-abelian.

Symmetric groups and set-theoretic operations
In the problems below, the symmetric group on a subset $$A \subseteq S$$ is understood as the subgroup of the symmetric group on $$S$$, comprising those permutations that fix every element outside $$A$$. The alternating group on $$A$$ comprises the even permutations that fix every element outside $$A$$.

Intersections of subsets

 * 1) Suppose $$A,B \subseteq S$$ are subsets. Prove that the symmetric group on $$A \cap B$$ is the intersection of the symmetric groups on $$A$$ and on $$B$$.

Unions of subsets
Suppose $$A$$ and $$B$$ are complements of each other in a set $$S$$.


 * 1) Prove that the symmetric groups on $$A$$ and $$B$$ intersect trivially as subgroups of the symmetric group on $$S$$. Further prove that every element of the symmetric group on $$A$$ commutes with every element of the symmetric group on $$B$$. Using this, show that the subgroup generated by the symmetric group on $$A$$ and the symmetric group on $$B$$ is isomorphic to the direct product of these groups. In particular, show that the symmetric group on $$S$$ contains a subgroup isomorphic to $$A \times B$$.
 * 2) Prove that the alternating groups on $$A$$ and $$B$$ intersect trivially as subgroups of the alternating group on $$S$$. Further prove that every element of the alternating group on $$A$$ commutes with every element of the alternating group on $$B$$. Using this, show that the subgroup generated by the alternating group on $$A$$ and the alternating group on $$B$$ is isomorphic to the direct product of these groups.
 * 3) NEEDS SOME THOUGHT: Suppose $$a_i, 1 \le i \le r$$ are nonnegative integers such that $$\sum_{i=1}^r a_i = n$$. Prove that the symmetric group on a set of size $$n$$ has a subgroup of order $$\prod_{i=1}^r a_i!$$.

Infinite sets

 * 1) Suppose $$G$$ is the symmetric group on an infinite set. Prove that $$G$$ contains a subgroup isomorphic to $$G \times \operatorname{Sym}(n)$$ for any natural number $$n$$.
 * 2) NEEDS SOME THOUGHT: Suppose $$G$$ is the symmetric group on an infinite set, and $$H$$ is any finite group. Prove that $$G$$ contains a subgroup isomorphic to $$G \times H$$.
 * 3) (This uses the fact that every infinite cardinal equals its double): Suppose $$G$$ is the symmetric group on an infinite set. Prove that $$G$$ contains a subgroup isomorphic to $$G \times G$$.
 * 4) NEEDS LOT OF THOUGHT: (breakdown on Cantor-Bernstein-Schroeder equivalent for groups): Give an example of two infinite groups $$A$$ and $$B$$ such that $$A$$ contains a subgroup isomorphic to $$B$$, $$B$$ contains a subgroup isomorphic to $$A$$, but $$A$$ and $$B$$ are not isomorphic.

Group actions on disjoint unions and on products

 * 1) Suppose $$G,H$$ are groups acting on the sets $$S,T$$ respectively. Construct a naturally induced action on $$G \times H$$ on the disjoint union $$S \sqcup T$$.
 * 2) Suppose $$G,H$$ are groups acting on the sets $$S,T$$ respectively. Construct a naturally induced action on $$G \times H$$ on the product $$S \times T$$.
 * 3) Using the previous problem, construct a homomorphism $$\operatorname{Sym}(m) \times \operatorname{Sym}(n) \to \operatorname{Sym}(mn)$$ where $$m,n$$ are natural numbers.

Group actions on power sets

 * 1) Suppose $$G$$ is a group acting on a set $$S$$. Construct a natural action of $$G$$ on the set of subsets of $$S$$. Prove that for this action, any two subsets in the same orbit have the same cardinality.
 * 2) NEEDS SOME THOUGHT: Consider, for the previous problem, the case where $$G = \operatorname{Sym}(S)$$ acting the usual way. Assume further that $$S$$ is finite. Prove that two subsets of $$S$$ are in the same orbit under the action if and only if they have the same cardinality.
 * 3) NEEDS SOME THOUGHT: Consider the case of a group $$G$$ acting on itself via left multiplication. Prove that the only subsets of $$G$$ that are fixed under the induced action on subsets are the empty set and the whole group.
 * 4) NEEDS SOME THOUGHT: Consider the case of a group $$G$$ acting on itself via left multiplication. Prove that the orbit of a subgroup $$H$$ of $$G$$ is precisely the set of left cosets of $$H$$ in $$G$$.

Group actions restricted to subgroups and composed with homomorphisms

 * 1) Suppose $$H$$ is a subgroup of $$G$$. Given an action of $$G$$ on a set $$S$$, construct an action of $$H$$ on $$S$$.
 * 2) Suppose $$\alpha:H \to G$$ is a homomorphism of groups. Given an action of $$G$$ on a set $$S$$, construct an action of $$H$$ on $$S$$ using $$\alpha$$ and the original action.
 * 3) Suppose $$H$$ is a subgroup of $$G$$. Consider the action of $$H$$ on $$G$$ by left multiplication. Prove thatthe orbit of any $$g$$ in $$G$$ is the right coset of $$H$$ in $$G$$ containing $$g$$.

Order and exponent
Recall that the order of an element is the order of the cyclic group it generates, and the exponent of a group is the least common multiple of the orders of all its elements.


 * 1) Prove that the order of a permutation equals the least common multiple of the sizes of all the cycles in its cycle decomposition.
 * 2) Prove that the exponent of the symmetric group on $$n$$ letters is the lcm of all the numbers from 1 to $$n$$.
 * 3) Prove that, for the symmetric group on three elements, the order equals its exponent, but there is no element whose order equals that exponent. Also, prove that for $$n \ge 4$$, the exponent of the symmetric group is always strictly smaller than the order.
 * 4) Prove that for $$n \ge 3$$, there does not exist any element in the symmetric group $$S_n$$, whose order equals the exponent of the group.

Cycle types
Recall that a transposition is a permutation that switches two elements and fixes all the remaining elements.


 * 1) Define $$F(k)$$ as the set of all permutations that can be expressed as the product of $$k$$ disjoint transpositions. Prove that the set of all permutations of order 2 is the disjoint union of the $$F(k)$$s for $$1 \le k \le [n/2]$$.
 * 2) NEEDS LOT OF THOUGHT: Find a formula for $$f(k) = |F(k)|$$, and show that it is a unimodal function of $$k$$: it first increases with $$k$$, and then decreases with $$k$$.
 * 3) NEEDS LOT OF THOUGHT: Prove that for $$n > 3$$, $$f(1) \ne f(k)$$ for any $$k > 1$$.

Abelian subgroups
A cycle or cyclic permutation is a permutation whose cycle decomposition has just one cycle. We say that two cycles are disjoint if they do not share an element.


 * 1) Prove that any two disjoint cycles permute.
 * 2) Consider a collection of disjoint cycles on $$n$$ elements of sizes $$m_1,m_2,\dots,m_r$$, such that $$\sum m_i = n$$. Prove that these cycles generate an abelian subgroup of order $$m_1m_2\dots m_r$$.
 * 3) NEEDS LOT OF THOUGHT: In the above problem, prove that the size of this Abelian subgroup is maximum when all the $$m_i$$ are either 2 or 3, with as many 3s as possible. Using this, prove that the size of this Abelian subgroup is bounded from above by $$3^{n/3}$$.
 * 4) In the symmetric group on four elements, prove that the double transpositions, along with the identity element, form a subgroup. (This gives an example of an Abelian subgroup not contained in any of the form described in problem (1)).