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