Tour:Examples peek three (beginners)

This page looks at concepts of subgroups, cosets, intersections and joins in a number of different contexts.

Real numbers and rational numbers

 * 1) Consider the group $$\mathbb{Q}$$ of rational numbers under addition and the subgroup $$\mathbb{Z}$$ of integers under addition. Prove that every coset of $$\mathbb{Z}$$ in $$\mathbb{Q}$$ has a unique representative in the interval $$[0,1)$$, and that this establishes a bijection between the rational numbers in $$[0,1)$$, and the coset space $$\mathbb{Q}/\mathbb{Z}$$ of $$\mathbb{Z}$$ in $$\mathbb{Q}$$.
 * 2) Consider the group $$\R$$ of reals under addition and the subgroup $$\mathbb{Z}$$ of integers under addition. Prove that every coset of $$\mathbb{Z}$$ in $$\mathbb{R}$$ has a unique representative in the interval $$[0,1)$$, and that this establishes a bijection between $$[0,1)$$ and the coset space $$\R/\mathbb{Z}$$.
 * 3) Prove that the coset space $$\R/\mathbb{Q}$$ is uncountable. Further, prove that every coset intersects every open interval $$(a,b)$$ in $$\mathbb{R}$$.

Monoids and Lagrange's theorem

 * 1) Consider the monoid of integers mod $$12$$ under multiplication. The elements of this monoid are the integers from 0 to 11, and the multiplication is defined as usual multiplication followed by taking the remainder modulo 12. Find submonoids of this monoid, whose cardinality is not a factor of 12. Thus, prove that the order of a submonoid may not divide the order of the monoid.
 * 2) Consider the following monoid: the elements of the monoid are the integers $$1,2,\dots,n$$, and we define the product of two elements as the larger of the elements. Prove that this is a monoid, and show that every subset containing the identity element is a submonoid. (Actually, any nonempty subset is a monoid under the induced multiplication, but the identity element of that subset need not be the same as the identity element of the whole monoid).

Vector spaces and subspaces

 * 1) Consider $$\R^2$$ as a two-dimensional vector space over $$\R$$. Prove that every line through the origin (i.e, every one-dimensional linear subspace) is a subgroup. Further, prove that the cosets of this subgroup are precisely the lines in $$\R^2$$ parallel to this line.
 * 2) In $$\R^3$$, prove that every plane through the origin (i.e., every two-dimensional subspace) is a subgroup, and its cosets are precisely the planes parallel to it. Similarly prove that every line through the origin is a subgroup, and its cosets are precisely the lines parallel to it.
 * 3) In $$\R^3$$, prove that either a plane and line intersect, or the line is parallel to a line on the plane.
 * 4) Prove that if a line is contained in a plane, then the line through the origin parallel to that line, is contained in the plane through the origin parallel to that plane.