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.

This page is a Examples peek page, part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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