HOMEWORKS
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Homework 1
For which fixed real numbers \(a, b, c\) the set of all real numbers subject
to operation \(x * y = ax + by + c\) forms a group?
Homework 2
Which of the following sets form a) group b) semigroup?
(i) \(\mathbb Z\) with respect to the subtraction \(a - b\);
(ii) \(\mathbb Z\) with respect to the operation \(a * b = a + b + ab\);
(iii) All subsets of a given set with respect to the symmetric difference
\(A \triangle B = (A \backslash B) \cup (B \backslash A)\) .
Homework 3
How many subgroups are there in \(S_3\)?
Homework 4
Find all homomorphisms from \(S_3\) to \(\mathbb Z / 2\mathbb Z\).
Homework 5
Prove that any nontrivial subgroup of \(\mathbb Z\) is isomorphic to
\(\mathbb Z\).
Homework 6
Which of the following groups are isomorphic: all real numbers subject to
addition, all nonzero real numbers subject to multiplication, all positive
real numbers subject to multiplication?
Homework 7
Is it true that:
1)
\(\mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z \simeq \mathbb Z/4\mathbb Z\)?
2)
\(\mathbb Z/2\mathbb Z \times \mathbb Z/3\mathbb Z \simeq \mathbb Z/6\mathbb Z\)?
Homework 8
Give an example of a group \(G\) and its subgroup \(H\) such that \(Z(G)\) is
properly contained in \(Z(H)\).
Homework 9
It is known that the map \(x \mapsto x^{-1}\) is an automorphism of a group.
What can be said about this group?
Homework 10
It is known that the automorphism group of a group \(G\) is trivial. What
can be said about \(G\)?
Homework 11
Is it true that the automorphism group of an abelian group is abelian?
Homework 12
Prove that if \(N \triangleleft G\) and \(M \triangleleft H\), then
\((N \times M) \triangleleft (G \times H)\). What can be said about the quotient
\((G \times H) / (N \times M)\)?
Homework 13
Determine whether the following groups are simple or not:
1) \(\mathbb Z/6\mathbb Z\).
2) \(\mathbb Z/p\mathbb Z\), where \(p\) is prime.
3) \(\mathbb Q\) with respect to addition.
4) \(\mathbb Q^*\) with respect to multiplication.
5) \(GL_n(\mathbb C)\).
6)
The group consisting of all \(3 \times 3\)
matrices of the form
\[
\left(\begin{matrix} 1 & a & b \\
0 & 1 & c \\
0 & 0 & 1
\end{matrix}\right)
\]
where \(a,b,c \in \mathbb Z\).
Homework 14
Prove that for an arbitrary group \(G\) the following maps are homomorphisms:
1) The map \(G \to S(G)\), where \(S(G)\) is the group of all bijections on the
set \(G\) subject composition, mapping \(x \in G\) to the map \(a \mapsto ax\).
2) The map \(G \to Aut(G\)), mapping \(x \in G\) to the inner automorphism
corresponding to \(x\) (i.e., \(a \mapsto x^{-1} a x\)).
What are the kernels of those homomorphisms? Apply the first homomorphism
theorem and draw conclusions.
Homework 15
Is the relation "to be a normal subgroup of a group" transitive?
Homework 16
Interpret the third homomorphism theorem for subgroups of \(\mathbb Z\) as some
divisibility property of integers.
Homework 17
Is it true that \(Z(G/Z(G))\) is trivial for an arbitrary group \(G\)?
Homework 18
Give examples of two rings consisting of 4 elements with non-isomorphic additive
groups.
Homework 19
Does the set of all subsets of a given set forms a ring subject to operations
of addition and multiplication defined as follows: \(A + B = A \triangle B\),
and
1) \(A \cdot B = A \cup B\);
2) \(A \cdot B = A \cap B\)?
Homework 20
Classify, up to isomorphism, algebras with unit of dimension 2 over the field
of complex numbers.
Homework 21
Show that the set \(K\) of matrices of the form
\[
\left(\begin{matrix} a & -b \\
b & a
\end{matrix}\right)
\]
where \(a,b \in GF(3)\) forms a subring in \(M_2(GF(3))\). Prove that \(K\) is
a field.
Homework 22
Let \(d\) be a nonzero square-free integer. Prove that the set
\(\mathbb Z[\sqrt{d}]\ = \{ a + b\sqrt{d} | a,b \in \mathbb Z\}\) is a
subring of \(\mathbb C\). Is this subring a subfield of \(\mathbb C\)? Is it
true that \(\mathbb Z[\sqrt{2}] \simeq \mathbb Z[\sqrt{3}]\)?
Homework 23
Let \(I\) and \(J\) be ideals of a algebra \(A\) such that \(A = I + J\). Prove
the isomorphism \(A / (I \cap J) \simeq A/I \oplus A/J\). Give an example of
such an algebra with nontrivial ideals \(I\) and \(J\).
Solution:
Define \(\varphi: A \to A/I \oplus A/J\) as \(a \mapsto (a+I,a+J)\). It is
obvious that \(\varphi\) is a homomorphism. \(Ker(\varphi)\) consists of
elements \(a \in A\) such that \(a+I = I\) and \(a+J = J\), what is equivalent
to \(a\in I\) and \(a\in J\), i.e., \(a\in I \cap J\). Let \((a+I,b+J)\) be an
arbitrary element in \(A/I \oplus A/J\). Since \(A = I+J\), we can write
\(a = i_1 + j_1\) and \(b = i_2 + j_2\) for some \(i_1,i_2 \in I\),
\(j_1,j_2 \in J\). Then
\[
\varphi(i_2 + j_1) = (i_2 + j_1 + I, i_2 + j_1 + J) = (j_1 + I, i_2 + J) =
(i_1 + j_1 + I, i_2 + j_2 + J) = (a+I,b+J) .
\]
This shows that \(\varphi\) is surjective. Now apply the first homomorphism
theorem.
Homework 24
Prove that the quaternion algebra over \(\mathbb R\) is a division algebra,
i.e., any nonzero element is invertible.
Created: Mon Feb 12 2024
Last modified: Mon May 6 16:57:58 CEST 2024