# Tour:Examples peek three (beginners)

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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 $\mathbb{R}$.

## Modular arithmetic

### 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.