Abstract algebra questions, solved and explained

Consider the following permutations in $S_{7}$ :
$α = (\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 6 & 1 & 7 & 4 & 2 & 5 & 3 \end{matrix})$
$β = (\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 4 & 3 & 1 & 7 & 2 & 5 & 6 \end{matrix})$
The order of $β$ is ?

Scoopedepalj 2022-02-28 Answered

Let G be a cyclic group of order 6. How many of its elements generate G?

Beverley Rahman 2022-02-28 Answered

Consider the incomplete character table for a group given below:
$\begin{array}{cc} (1) & (1) & (2) & (2) & (2) \\ 1 & a & b & c & d \\ x_{1} & 1 & 1 & 1 & 1 & 1 \\ x_{2} & 1 & 1 & - 1 & - 1 & 1 \\ x_{3} & 1 & 1 & - 1 & - 1 & - 1 \\ x_{4} & 2 & - 2 & 0 & 0 & 0 \end{array}$
All the conjugacy classes are there.
1) What is the order of the group?
2) How many characters are missing?
3) Find the missing character and complete the table.
4) Find the order of the Kemel of the missing character.

sacateundisco8i3 2022-02-28 Answered

Express as the product of disjoint cycles
a) (123)(45)(16789)(15)
b) (12)(123)(12)

asistioacer 2022-02-24

There are 2 photos inserted the one is the part 1 of the problem and the other is the part 2 . The picture with the number 1 is the part 1 and the x and y axis graph pic is part 2.

asistioacer 2022-02-24

There are 2 photos inserted the one is the part 1 of the problem and the other is the part 2 . The picture with the number 1 is the part 1 and the x and y axis graph pic is part 2.

Kamari Simon 2022-02-15 Answered

Abelian group order divides 1cm
I want to prove the following:
Let G be an abelian group and let $a_{1}, \dots, a_{r} \in G$ then $| a_{1} \dots a_{r} |$ (the order of $a_{1} \dots a_{r}$ ) divides $1 c m (| a_{1} |, \dots, | a_{r} |)$
So far I have shown that ${(a_{1}, \dots, a_{r})}^{1 c m (| a_{1} |, \dots, | a_{r} |)} = 1$ but I dont

Bryant Miranda 2022-02-15 Answered

if $f : R \to R$ is a homomorphism and p is an integer-coefficient polynomial, then:
$f (p (r)) = p (f (r)) \forall r \in R$

kaliitcri 2022-02-15 Answered

Northcott Multilinear Algebra Universal Property Proof
Northcott Multilinear Algebra poses a problem. Consider R-modules $M_{1}, \dots, M_{p}$ , M and N. Consider multilinear mapping
$ψ : M_{1} \times \dots \times M_{p} \to N$
Northcott calls the universal problem as the problem to find M and multilinear mapping $ϕ : M_{1} \times \dots \times M_{p} \to M$ such that there is exactly one R-module homomorphism $h : M \to N$ such that $h \circ ϕ = ψ$ .
If $λ and λ^{'}$ exist I understand why the equalities at the end of the sentence follow, based on the satisfaction of the universal problem. I can't see however why homomorphisms $λ and λ^{'}$ should exist.
I did more group theory many years ago and this is my first serious foray into "modules" so I wouldn't be surprised if there is something obvious I'm missing.
my thoughts: Clearly M and M′ are both homomorphic to N through h and h′, I'm not sure if this says anything about a relationship between M and M′ though.
If h′ were injective I could say something like $λ (m) = h^{^{'} - 1} (h (m))$ but I don't know if there is any guarantee that h′ is injective..
Likewise, if $ϕ$ were injective I could define $λ (m) = ϕ^{'} (ϕ^{- 1} (m))$ but again I don't know why this would be the case...
I've tried replacing M and N with more familiar vector spaces and R-module homomorphisms by multilinear maps for better intuition but no luck.. I do know that if M and M′ are vector spaces with the same dimension then there is an isomorphism between them. I guess more generally if M and M′ have different dimensions (say $\dim (M^{'}) > \dim (M))$ then there is a homomorphism from M into a subspace of M′ and another homormophism from M′ onto M. Maybe this carries over to modules and is in the right direction for what I need...?

Alaina Ortiz 2022-02-15 Answered

Module with matrix multiplication
Define the rational $3 \times 3$ -matrix
$A := (\begin{matrix} 1 & 4 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 2 \end{matrix})$
We then define a $Q [X]$ -module on $V = Q^{3}$ such that
$Q [X] \times V \to V, (P, v) \to P (A) \cdot v$
where $P (A) \in Q^{3 \times 3}$ is achieved by plugging in matrix A into polynomial P and $P (A) \cdot v$ is therefore the matrix-vector-multiplication. Call this module $V_{A}$ .
Argue if $V_{A} = Q [X] (\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}) \oplus Q [X] (\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix})$ .

Jacob Stein 2022-02-15 Answered

Convoluted definition of a set: $H (S) = {x \in G ∣ \exists n \in N, \exists {x_{1}, x_{2}, \dots, x_{n}} \subseteq S \cup S^{- 1}, x = x_{1} \dots x_{n}}$ .

Krystal Villanueva 2022-02-15 Answered

Very basic question about localizations
Let A be a ring, S a multiplicatively closed subset. Is it true that $\frac{a}{1} \in S^{- 1} A$ is a non zero-divisor if and only if $a \in A$ is?

Quinten Crawford 2022-02-15 Answered

Let $G = a$ . The smallest subgroup of G containing $a^{8} and a^{12}$ is generated by
1)
2)
3)
4)
Arithmetical subtraction (-) is binary relation on
1) N
2) Z+
3) Q
4) Z-

pimpinan15t 2022-02-14 Answered

If H and K are subgroups of G, $| H | = 20 and | K | = 32$ then a possible value of $| H \cap K |$ is
1) 2
2) 8
3) 6
4) 16

Aryan Phillips 2022-02-14 Answered

Fundamental question about field extensions
Let k be an algebraically closed field. Let $K \subset k$ be a subfield, and suppose that it has an algebraic extension $K \subset L$ . Is it true (or at least well defined, because I'm not even sure about this) that $L \subseteq k$ ? One slightly different point of view is: given injections $i : K \to k, j : K \to L$ , the last one being algebraic, is there an injection $l : L \to k$ such that $l \circ j = i$ (i.e. l is a homomorphism of K-algebras)?

Amiya Arellano 2022-02-14 Answered

Is there a name for the set $E (x, n) = {x^{p} ∣ p \in N \land \leq p \leq n - 1} with x \in G, G$ a group?

Seelakant6vr 2022-02-14 Answered

Simple question about finitely generated algebras
I have two finitely generated algebras A and B over a field K such that $B \subseteq A$ . Is it true that $A = B [a_{1}, \dots, a_{n}]$ for some $a_{1}, \dots, a_{n} \in A$ ?

uheaeb56e 2022-02-14 Answered

Consider the following permutations in $S_{9}$ :
$σ = (\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 3 & 5 & 2 & 1 & 4 & 6 & 9 & 7 & 8 \end{matrix})$ and
$τ = (\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 5 & 2 & 6 & 7 & 1 & 3 & 8 & 9 & 4 \end{matrix})$
Find $α$ by solving $σ α = τ$ . Write your answer in cycle notation.

squaloideyjj 2022-02-14 Answered

Julio is studying for his final exams in chemistry and algebra. He knows he only has 12 hours to study and it will take him at least 2 more hours to study for algebra than chemistry. The following system of equations represents the situation, where x represents algebra and y represents chemistry:
$x + y \leq 12$
$x \geq y + 2$
Which of the following can Julio do?
Study algebra for 7 hours and chemistry for 5 hours
Study algebra for 8 hours and chemistry for 4 hours
Study algebra for 7 or 8 hours and chemistry for 4 or 5 hours
None of the above

Randy Mejia 2022-02-14 Answered

The number of elements in $S_{6}$ is
1) 6
2) 720
3) 36
4) 360

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Abstract algebra questions and answers