HOMEWORKS
Homework 2
Find all groups which have exactly
1) one subgroup;
2) two subgroups;
3) three subgroups.
Homework 9
Prove that \([[S_3,S_3],[S_3,S_3]]\) is trivial.
Homework 10
Prove that the commutator of the group consisting of all \(3 \times 3\)
matrices of the form
\[
\left(\begin{matrix} 1 & a & b \\
0 & 1 & c \\
0 & 0 & 1
\end{matrix}\right)
\]
with \(a,b,c \in \mathbb Z\), is equal to its center.
Homework 11
Describe up to isomorphism all rings consisting of 3 elements.
Homework 14
Give an example of a noncommutative ring with a nontrivial commutative quotient.
Homework 16
Prove that if \(I, J\) are ideals in a ring \(R\), then \(I + J\) is an ideal in
\(R\).
Homework 17
Let \(I\) be an ideal in an algebra \(A\), and \(J\) an ideal in an algebra
\(B\). Prove that \(I \otimes B\) and \(A \otimes J\) are ideals in
\(A \otimes B\). Describe the quotient
\((A \otimes B)/(I \otimes B + A \otimes J)\).
Homework 18
Prove that the quaternion algebra over an arbitrary field is simple.
Solution.
Created: Tue Feb 22 2022
Last modified: Mon Apr 22 22:11:28 CEST 2024