Tour:Interdisciplinary problems three (beginners)

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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. Define the closed subgroup generated by a subset $S$ in a topological group $G$ to be the intersection of all closed subsets 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 $S$.
4. (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 $G=H_n$.

Combinatorics

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 $f_1$.

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)
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