Tour:Interdisciplinary problems three (beginners)

The problems here relate ideas in group theory to ideas in other subjects. Learners who already have some experience with the other subjects may try these problems to cement their understanding of what has been learned in parts one and two of the tour. Moreover, these problems build on themes introduced in Tour:Interdisciplinary problems two (beginners), so a review of those problems may be helpful.

Topology
A topological group is a set $$G$$ with a topology as well as a group structure, so that the inverse map is continuous with respect to the topology, and the multiplication map is continuous as a map from $$G \times G$$ to $$G$$.

We'll need to remember the following facts about topological groups:


 * For a topological group $$G$$ and an element $$g \in G$$, the left multiplication by $$g$$, given by $$x \mapsto gx$$, is a homeomorphism. It sends open subsets to open subsets and closed subsets to closed subsets. Similarly, right multiplication by $$g$$ is a homeomorphism.
 * An arbitrary union of open subsets of a topological space is open.
 * A finite intersection of open subsets of a topological space is open.
 * The subset of a topological space is closed if and only if its complement is open.

Cosets, intersections, open and closed

 * 1) Prove that if a subgroup of a group is open (resp., closed) so is every left and right coset of the subgroup.
 * 2) Prove that any open subgroup of a topological group is closed.
 * 3) Prove that any closed subgroup of finite index in a topological group is open.
 * 4) Prove that if $$H \le K \le G$$ are groups, with $$G$$ a topological group, and $$H$$ open in $$G$$, then $$K$$ is open in $$G$$.
 * 5) Prove that if $$H \le K \le G$$ are groups, with $$G$$ a topological group, $$[K:H]$$ finite, and $$H$$ closed in $$G$$, then $$K$$ is closed in $$G$$.
 * 6) (For those who know connectedness) Prove that in a connected topological group, there is no proper open subgroup, and every proper closed subgroup has infinite index.
 * 7) (For those who know compactness) Prove that in a compact topological group, every open subgroup has finite index.
 * 8) Prove that an arbitrary intersection of closed subgroups is closed.

Subgroup generated

 * 1) NEEDS SOME THOUGHT: If $$U$$ is an open subset of $$G$$, prove that the subgroup generated by $$U$$ is an open subgroup of $$G$$
 * 2) Use the previous problem and an earlier exercise to deduce that if $$G$$ is a connected group, it is generated by any nonempty open subset.
 * 3) NEEDS SOME THOUGHT: Prove that the closure (in a topological sense) of a subgroup of a group must also be a subgroup.
 * 4) NEEDS SOME THOUHT: Define the closed subgroup generated by a subset $$S$$ in a topological group $$G$$ to be the intersection of all closed subgroups of $$G$$ containing $$S$$. Prove that the closed subgroup generated by $$S$$ is the same as the closure (in a topological sense) of the subgroup generated by $$S$$.
 * 5) (For those who have seen groups of motions in Euclidean space) NEEDS LOT OF THOUGHT: Prove that a join of two closed subgroups of a topological group need not be closed. (An explicit example can be constructed using reflections about lines at an irrational angle)

Measure theory
A left-invariant measure $$\mu$$ on a group $$G$$ is a measure with the property that for any $$g \in G$$, the map $$x \mapsto gx$$ is a measure-preserving transformation. In other words, for any measurable subset $$S$$ of $$G$$, $$gS$$ is measurable and $$\mu(gS) = \mu(S)$$.

In the exercises, we shall assume that $$G$$ is a group with left-invariant measure $$\mu$$, and we will further assume that $$\mu(G)$$ is finite.

Lagrange's theorem and applications

 * 1) Prove that if $$H$$ is a measurable subgroup of $$G$$, then $$\mu(H)[G:H] = \mu(G)$$. In other words, the index of $$H$$ in $$G$$ equals the ratio of the measure of $$G$$ to the measure of $$H$$.
 * 2) Prove that if $$H$$ is a proper measurable subgroup of $$G$$, then $$\mu(H) \le \frac{1}{2} \mu(G)$$.
 * 3) NEEDS SOME THOUGHT: Prove that if we have an ascending chain of subgroups of $$G$$:

$$H_0 \le H_1 \le H_2 \le \dots $$

where each $$H_i$$ is measurable, and $$G$$ is the union of the $$H_i$$s, then there exists some $$n$$ for which $$G = H_n$$.

Quasigroups

 * 1) Prove that if $$H$$ is a proper subquasigroup of a finite quasigroup, then the size of $$H$$ is at most half the size of $$G$$. (Hint: Use the fact that multiplying any element outside $$H$$ with any element inside $$H$$ gives an element outside $$H$$.)
 * 2) Prove that if $$H$$ is a proper subquasigroup of an infinite quasigroup $$G$$, then the complement $$G \setminus H$$ is infinite.

Sidon subsets

 * 1) A subset $$S$$ of an Abelian group $$G$$ is termed a Sidon subset (we studied these in interdisciplinary problems two). A maximal Sidon subset is a Sidon subset not properly contained in a bigger Sidon subset. Prove that a maximal Sidon subset cannot completely be contained inside a coset of a proper subgroup.

Fibonacci sequences

 * 1) A Fibonacci sequence in an Abelian group $$G$$ is defined as follows. $$f_0$$ and $$f_1$$ are defined as arbitrary elements of $$G$$, and $$f_n$$ is defined inductively as $$f_{n-1} + f_{n-2}$$. Prove that any Fibonacci sequence in $$G$$ lies inside the subgroup generated by $$f_0$$ and $$f_1$$.

Number of generators needed to express a law
The axiomatization for groups includes associativity, that needs universal quantification over three elements. Can we get an axiomatization that does not require quantification over three elements? We explore this and related questions here.


 * 1) Suppose a variety of algebras can be described using a collection of universal identities where no identity requires universal quantification over more than $$n$$ elements. Prove that an algebra with the required operations is in the variety if and only if for any $$n$$ elements of the algebra, there is a subalgebra of the algebra containing those $$n$$ elements, that is in the variety.
 * 2) NEEDS LOT OF THOUGHT: Construct a magma that is not a semigroup, but with the property that the submagma generated by any two elements is a semigroup. Using this, show that the variety of semigroups cannot be axiomatized with laws having fewer than three universally quantified elements.
 * 3) NEEDS LOT OF THOUGHT: Construct an algebra loop that is not a group, but with the property that the subloop generated by any two elements is a subgroup. Using this, show that the variety of groups cannot be axiomatized with laws having fewer than three universally quantified elements.

Permutational invariance of laws
Given a binary operation, we can consider the opposite binary operation: a binary operation that acts on the inputs in the reverse. $$*$$ and $$\cdot$$ are opposite if $$x * y = y \cdot x$$ for all $$x$$ and $$y$$.


 * 1) Prove that a binary operation is commutative if and only if it equals the opposite operation.
 * 2) Prove that a binary operation is associative if and only if the opposite operation is associative.
 * 3) Prove that a binary operation admits a two-sided neutral element if and only if the opposite operation does, and that the neutral element is the same for both. (Note: In fact, left neutral elements for a binary operation are right neutral for the opposite operation).
 * 4) Prove that a binary operation is left-cancellative if and only if the opposite operation is right-cancellative.
 * 5) Prove that an element is left-invertible for a binary operation if and only if it is right-invertible for the opposite binary operation.

In other words, all the laws we have used are preserved on passing to the opposite operation. Thus:


 * 1) Prove that a magma is a semigroup if and only if its opposite magma is.
 * 2) Prove that a magma is a monoid if and only if its opposite magma is.
 * 3) Prove that a magma is a group if and only if its opposite magma is.
 * 4) Prove that a magma is a quasigroup if and only if its opposite magma is.