Cosets Homework
Directions: Write your homework legibly with a pencil.
Complete the homework on time and by yourself.
For each problem, write the instructions, label the solution/proof and show all steps.
Do not turn in your scratch work.
Staple your pages together, with the problems in the correct order, and use this page as a cover sheet with your name at the top right.
(1) Determine the right cosets of
in
(2) Determine the right cosets of
in
(3) Determine the right cosets of
in
(4) Determine the right cosets of
in
(5) Verify that if
then
is a subgroup of
(with
as the operation in each component).
Describe the right cosets of
in
geometrically.
(6) Prove that if
is a subgroup of a group
and
then the following conditions are equivalent: (i)
(ii)
for some
(iii)
and (iv)
(7) Verify that if
is a subgroup of an Abelian group
and
then the right coset of
to which
belongs is the same as the left coset of
to which
belongs.
(8) Let
and
Determine the right cosets and the left cosets of
in
Then verify that the collection of right cosets is different from the collection of left cosets.
(9) Prove that each right coset of
in
contains precisely one element from
(10) Prove that a subset
of a group
cannot be a right coset of two different subgroups of
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Cite this as: Cosets Homework Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/cosets-homework.html
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