HOMEWORKS
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Homework 1
For which fixed real numbers \(a, b, c\) the set of all real numbers subject
to the 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
Prove that the group \(\mathbb R^*\) (the multiplicative group of \(\mathbb R\))
is not isomorphic to its subgroup consisting of all positive numbers.
Homework 7
It is known that the automorphism group of a group \(G\) is trivial. What
can be said about \(G\)?
Homework 8
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 9
Give an example of a group \(G\) and its subgroup \(H\) such that \(Z(G)\) is
properly contained in \(Z(H)\).
Homework 10
Let \(H\) be a subgroup of index 3 in a group \(G\) such that at least for one
element \(x \in G\) not belonging to \(H\), the respective left and right cosets
coincide: \(xH = Hx\). Prove that \(H\) is a normal subgroup. Provide example of
such \(G\) and \(H\).
Homework 11
Determine whether the following groups are simple or not:
1) \(\mathbb Z/6\mathbb Z\).
2) \(\mathbb Q\) with respect to addition.
3)
\(SL_n(\mathbb C)\).
4)
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 12
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 13
Interpret the third homomorphism theorem for subgroups of \(\mathbb Z\) as a
certain divisibility property of integers.
Homework 14
Is it true that \(Z(G/Z(G))\) is trivial for an arbitrary group \(G\)?
Homework 15
Prove that the commutator of the group from Homework 11, part 4, is equal to its
center.
Homework 16
Describe up to isomorphism all rings consisting of 3 elements.
Homework 17
Give an example of two rings consisting of 4 elements with non-isomorphic
additive groups.
Homework 18
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 19
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 proper 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 20
Prove that the quaternion algebra over \(\mathbb R\) is a division algebra,
i.e., any nonzero element is invertible.
Homework 21
Prove that the quaternion algebra over an arbitrary field is simple.
Solution:
Homework 22
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 23
Is it possibe to represent the quaternion algebra as a nontrivial tensor product
of two algebras (i.e., neither of the tensor factors is isomorphic to the ground
field)?
If the question looks difficult, try to narrow it down by introducing additional
(reasonable) assumptions (assumption on the ground field, assumption on the
tensor factors, etc.).
Homeworks due Tuesday April 22
Homework 24
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 25
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 26
In what follows, we will use the so-called evaluation map \(K[x] \to K\) mapping
a polynomial \(f(x)\) with coefficients in the field \(K\) to \(f(0)\). This is
obviously a surjective homomorphism of rings.
Theorem. Any field is prime.
Proof. Let \(F\) be a field, and \(K\) is the prime subfield of \(F\).
Suppose \(F \ne K\), choose an element \(\alpha \in F \backslash K\), and
consider the field extension \(K \subset K(\alpha)\). The latter extension is
either algebraic or transcendental.
If it is algebraic, then \(K(\alpha) = K[\alpha]\) (i.e., any element of
\(K(\alpha)\) is a polynom of \(\alpha\) with coefficients in \(K\)). Consider
the evlauation map \(K[\alpha] \to K\).
If the extension is transcendental, then extend the evaluation map to the map
\(K(\alpha) \to K\) mapping a formal fraction \(f(\alpha) / g(\alpha)\) to
\(f(0) / g(0)\).
In both cases, by the first homomorphism theorem, \(K(\alpha) / I \simeq K\),
where \(I\) is the kernel of the respective evaluation map. But as we proved in
the class, each field is simple as a ring, hence either \(I = K(\alpha)\) and
\(K\) is isomorphic to zero ring, a contradiction, or \(I = 0\), hence
\(K(\alpha) = K\), what contradicts the choice of \(\alpha\). This shows that
\(F = K\). \(\blacksquare\)
This is absurd, as we know a lot of non-prime fields. Where is an error?
Created: Mon Feb 12 2024
Last modified: Tue Apr 8 2025 18:53:04 CEST