HOMEWORKS
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Scores
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 \(G\)?
Homeworks due Tuesday March 4
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 \(H\).
Homeworks due Tuesday March 11
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.
Created: Mon Feb 12 2024
Last modified: Tue Mar 4 2025 18:57:59 CET