Free modules are projective, and projective modules are direct summands of free modules. Are there examples of projective modules that are not free? (I know this is not possible for modules of fields.) Free modules are torsion-free. But is the inverse true? Are there examples of torsion-free modules that are not free? Thank you~

As the title suggests. Let $G$ be a group, and suppose the function $\phi: G \to G$ with $\phi(g)=g^3$ for $g \in G$ is a homomorphism. Show that if $3 \nmid |G|$, $G$ must be abelian. By considering $\ker(\phi)$ and Lagrange’s Theorem, we have $\phi$ must be an isomorphism (right?), but I’m not really sure […]

Suppose you know the following two things about a group $G$ with $n$ elements: the order of each of the $n$ elements in $G$; $G$ is uniquely determined by the orders in (1). Question: How difficult is it to recover the group structure of $G$? In other words, what is the best way to use […]

If $$R=\left\{ \begin{pmatrix} a &b\\ 0 & c \end{pmatrix} \ : \ a \in \mathbb{Z}, \ b,c \in \mathbb{Q}\right\} $$ under usual addition and multiplication, then what are the left and right ideals of $R$?

I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood everything until we covered the Yoneda Lemma, after which point I lost interest. I guess what I’m asking for are some concrete examples […]

If we let $S_n$ denote the symmetric group on $n$ letters, then any element in $S_n$ can be written as the product of disjoint cycles, and for $k$ disjoint cycles, $\sigma_1,\sigma_2,\ldots,\sigma_k$, we have that $|\sigma_1\sigma_2\ldots\sigma_k|=\operatorname{lcm}(\sigma_1,\sigma_2,\ldots,\sigma_k)$. So to find the maximum order of an element in $S_n$, we need to maximize $\operatorname{lcm}(\sigma_1,\sigma_2,\ldots,\sigma_k)$ given that $\sum_{i=1}^k{|\sigma_i|}=n$. So […]

Given an exact sequence of vector spaces $$\cdots\longrightarrow V_{i-1}\longrightarrow V_{i}\longrightarrow V_{i+1}\longrightarrow\cdots$$ I want to show that it is the same as having a collection of short ones such that $$0\longrightarrow K_i \longrightarrow V_i \longrightarrow K_{i+1}\longrightarrow0$$ So to start I want to show exactness at an arbitrary $V_i$, so I space them suggestively: $$\begin{array}{c} 0&\rightarrow &K_{i-1}&\rightarrow […]

Let $A$ be a commutative ring, $I$ an ideal of $A$ and $M$ an module over $A$. Is it true that $\operatorname{Ann}(M/IM) = \operatorname{Ann}(M) + I$? One inclusion is certainly true, but I don’t know about the other one. If the above statement does not hold in general, does it perhaps for finitely generated modules?

I’m asked to show that $G=GL_2(\Bbb Z/2\Bbb Z)$ is isomorphic to $S_3$. I have few ideas but I don’t manage to put them all together in order to obtain a satisying answer. I first tried using Cayley’s theorem ($G$ is isomorphic to a subgroup of $S_6$), and I also noticed that $\operatorname{Card}(G)=\operatorname{Card}(S3)=6$ & that they’re […]

$\zeta_{15}$ is $15$th primitive $n$th root of unity. Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$ I know that if $p$ is prime then $G=Gal(\mathbb{Q}(\zeta_{p})/\mathbb{Q})=\mathbb{Z_{p}}^*$ but when $p$ is not prime, I am not show how to solve this if not prime. Does my required group have any relation to some cyclic group $\mathbb{Z_n}$, i.e. […]

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