Abstract algebra questions, solved and explained

Recent questions in Abstract algebra

et3atissb 2022-04-23 Answered

Is k[[x]] ever a finitely generated k[x](x) module?
For k a field, the localization $k {[x]}_{(x)}$ naturally includes into k[[x]]. I can prove that if $k = C,$ , then this inclusion is not surjective, and k[[x]] is not even finitely generated over $k {[x]}_{(x)}$ , because $C {[x]}_{(x)}$ corresponds to rational functions holomorphic at 0, and $C [[x]]$ corresponds to germs of all functions holomorphic at 0, and so with some complex analysis you can see the finite generation is impossible. But over an arbitrary field, is this claim still true?

windpipe33u 2022-04-23 Answered

Show that the Lorentz boosts,
$(\begin{matrix} γ & - β γ \\ - β γ & γ \end{matrix})$
form a one-parameter Lie group.

coraletsmmh 2022-04-23 Answered

Product of two cyclic groups is cyclic iff their orders are co-'
Say you have two groups $G = ⟨ g ⟩$ with order n and $H = ⟨ h ⟩$ with order m. Then the product $G \times H$ is a cyclic group if and only if $gcd (n, m) = 1$

Zack Wise 2022-04-22 Answered

Notation of homeomorphism from B(H) to B(K), corresponding to unitary transformation of Hilbert spaces
Let U be a unitary transformation from Hilbert space H to Hilbert space K.
How do you call a *-homomorphism f from B(H) to B(K), defined by $f (a) = U a U^{- 1}$ ?
I'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech.

Saige Shannon 2022-04-22 Answered

How does cancellation work in polynomial quotient rings?
$\frac{Z [x]}{(2 x - 6, 6 x - 15)}$
Can I just automatically say that
$\frac{Z [x]}{(2 x - 6, 6 x - 15)} \tilde{=} \frac{Z [x]}{(x - 3, 6 x - 15)}$
by just dividng the first polynomial by 2?

Dereon Guzman 2022-04-21 Answered

If the cardinality of a ring R is greater than 1, then $1 \neq 0$ .

devojciq5o 2022-04-20 Answered

Solving (quadratic) equations of iterated functions, such as
$f (f (x)) = f (x) + x$

Jaylyn Villarreal 2022-04-17 Answered

We want to prove that $(Z_{7}, \oplus)$ is group. I have difficulty proving associativity axiom. The solution reads
Let $a \in Z_{7}, b \in Z_{7}$ and $c \in Z_{7}$ . By Theorem 3.4.10 we only need to show
$(a + (b + c)) b mod 7 = ((a + b) + c) b mod 7 .$
This holds since $a + (b + c) = (a + b) + c$ for all integers a, b, and c by the associative property of the integers. Hence $\oplus$ is associative.
Therorem 3.4.10. Let a and b be integers, and let m be a natural number. Then
$(a + b) mod m = ((a mod m) + (b mod m)) mod m$

Kendal Day 2022-04-16 Answered

Multiples of 4 as sum or difference of 2 squares
Is it true that for any $n \in N$ we can have $4 n = x^{2} + y^{2}$ or $4 n = x^{2} - y^{2}$ , for $x, y \in N \cup (0)$ ?
I was just working out a proof and this turns out to be true from $n = 1 to n = 20$ . After that I didn't try, but I would like to see if a counterexample exists for a greater value of n.

Essence Ingram 2022-04-16 Answered

We were learning about centralizers in my abstract algebra class, and I had a thought about centralizers.
Is every subgroup of a group G the centralizer of some element? And if not, how many of the subgroups are?

gabolzm6d 2022-04-16 Answered

What are all the homomorphisms between the rings $Z_{18} and Z_{15}$ ?
Any homomorphism $ψ$ between the rings $Z_{18} and Z_{15}$ is completely defined by $ψ (1)$ . So from
$0 = φ (0) = φ (18) = φ (18 \cdot 1) = 18 \cdot φ (1) = 15 \cdot φ (1) + 3 \cdot φ (1) = 3 \cdot φ (1)$
We get that $ψ (1)$ is either 5 or 10. But how can I prove or disprove that these two are valid homomorphisms?

Papierskiix5n 2022-04-15 Answered

Let A be a discrete valuation ring, and let $a \in A$ be a non-zero element. Compute the integral closure of $B : = A [X, \sqrt{a X}]$

Santos Mooney 2022-04-14 Answered

Let G be a finite group. Let T be an element of Aut(G) such that
$T^{2} = I$ , and $x T = x \Leftrightarrow x = e$ .
Then, G is abelian.

Breanna Mcclure 2022-04-14 Answered

Is there a particular characterization of $A u t (Q^{3})$ ?

Bradley Barron 2022-04-14 Answered

Find the isomorphic ring with
$Z_{6} \frac{x}{⟨ x^{3} - x ⟩}$
(Chinese remainder thm)

Rubi Riggs 2022-04-13 Answered

Let G be a finite group with $c a r d (G) = p^{2} q$ with $p < q$ two ' numbers. We denote $s_{q}$ the number of q-Sylow subgroups of G and similarly for p. I have just shown that $s_{q} \in {1, p^{2}}$ . Now I want to show that
$\underset{S \in S y l_{q} (G)}{\cup} S ∖ {1} = {\dot{\cup}}_{S \in S y l_{q} (G)} S ∖ {1}$
i.e. that for $S, T \in S y l_{q} (G)$ with $S \neq T$ we that $S ∖ {1} \cap T ∖ {1} = \emptyset$

Ansley Sparks 2022-04-13 Answered

Jacobson radical of the ring of lower triangular $n \times n$ matrices over $Z$

Rosa Townsend 2022-04-13 Answered

Isomorphism between quotient of ring and principal ideals
Let R be a principal ideal domain and suppose that $u, u^{'} \in R$ are such that $\frac{R}{(u)} \tilde{=} \frac{R}{(u^{'})}$ as R-modules, where (u) denotes the ideal generated by u. Is it true that $u = α u^{'}$ , where $α \in R$ is invertible?

Ashleigh Mitchell 2022-04-13 Answered

Is there any relation between ${Sym}_{n}$ and ${Sym}_{k}$ where $k < n$ ? Is ${Sym}_{k} \subset {Sym}_{n}$ ? Since $k! ∣ n! \forall k < n$ they satisfy Lagrange's theorem, but of course that doesnt guarantee the existence of a subgroup

Devin Dougherty 2022-04-12 Answered

How do I prove that this ideal is not a ' ideal?
Let K be a field and $R = \frac{K [X, Y]}{(X Y)} . F or P \in K [X, Y]$ we denote $[P]$ its class in R. Show that the Ideal (XY) is not a ' ideal.

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