# Tour:Examples peek three (beginners)

From Groupprops

This page is a Examples peek page, part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Interdisciplinary problems three (beginners)|UP: Introduction three |NEXT: Introduction four (beginners)PREVIOUS SECTION Examples peek: Examples peek two|NEXT SECTION Examples peek: Examples peek four

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

## Contents

## Real numbers and rational numbers

- Consider the group of rational numbers under addition and the subgroup of integers under addition.
**Prove that**every coset of in has a unique representative in the interval , and that this establishes a bijection between the rational numbers in , and the coset space of in . - Consider the group of reals under addition and the subgroup of integers under addition.
**Prove that**every coset of in has a unique representative in the interval , and that this establishes a bijection between and the coset space . -
**Prove that**the coset space is uncountable. Further,**prove that**every coset intersects every open interval in .

## Modular arithmetic

### Monoids and Lagrange's theorem

- Consider the monoid of integers mod 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. - Consider the following monoid: the elements of the monoid are the integers , 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

- Consider as a two-dimensional vector space over .
**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 parallel to this line. - In ,
**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. - In ,
**prove that**either a plane and line intersect, or the line is parallel to a line on the plane. -
**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)PREVIOUS: Interdisciplinary problems three (beginners)|UP: Introduction three |NEXT: Introduction four (beginners)PREVIOUS SECTION Examples peek: Examples peek two|NEXT SECTION Examples peek: Examples peek four

General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part