# Questions tagged [associativity]

This is the property shared by many binary operations including group operations. For a binary operation $\cdot$, associativity holds if $(x\cdot y)\cdot z = x \cdot(y\cdot z)$.

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### Proving that a group homomorphism preserves associativity

I felt this was trivial, but I wanted to make sure. The proofs I've read which show that the image of a group homomorphism is a subgroup of its codomain only prove that closure, identity, and inverse ...
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### Speculative- Associativity and Information Loss

In Axler's Linear Algebra Done Right, the following problem (1.B.6) was posed: Let $\infty$ and $-\infty$ denote two distinct objects, neither of which is in $\mathbb{R}$. Define an addition and ...
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### Associativity of a semidirect product

I have the following problem. Let $$0\to A\to G\to Q\to 1$$ be a central group extension with $A$ abelian. Assume that this extension splits, i.e., $G\cong A\rtimes Q$. Now consider an action of ...
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### Why is it that $\frac{d}{dx}((a+2bx)e^{3x}) \neq \frac{d}{dx}((ae^{3x}+2bxe^{3x}))$

I was computing a differential equation and I ended up with the following result: $\frac{d}{dx}((a+2bx)e^{3x})$. My intuition was to distribute $e^{3x}$, however I got the wrong answer. When I wrote ...
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### Prove associative property of $\Bbb Z [x]$ ($\Bbb Z [x]$ is the set of all polynomials with variable x and integer coefficients)

I can't seem to find the answer anywhere to make sure the steps I did are right or wrong, and excuse me if the question is kinda dumb :") I want to prove that $\Bbb Z [x]$ is a ring, so I want to ...
1 vote
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### How to prove that (R*, x) is associative?

I understand that (R*, x) (under multiplication) is a group, where R* is the real numbers excluding 0. It has an identity element (1) Has an inverse (1/x for all x in the group) but I am unsure how to ...
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### If $(\mathbb Z,*)$ is invertible and has cancellation property, does it imply that is associative? [closed]

If $(\mathbb Z,*)$ is invertible and has cancellation property, does it imply that is associative? I know that if is invertible and has associative, then is a group and has cancellation property. But ...
1 vote
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### What is the correct reading of $A+B+C$?

We have many operations where when we write it in a form that does not make it explicit which expressions are the inputs to which operands, for example $1-1-1$ is a form. Is there a name for this kind ... 86 views

### Is every associative $n$-ary operation with an identity element induced by a monoid?

Given any $n$-ary operation $*$ on a set $X$, an identity element for $*$ is an element $e \in X$ such that $x*e*e*...*e=e*x*e*e*...*e=...=e*e*...*e*x=x$ ($n-1$ $e$s in each product) for all $x \in X$....
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### Proof of Matrix Cross-Multiplication Distributive Property

I am trying to prove that $A\times(B+C)=A \times B + A \times C.$ I have managed to prove this by finding the LHS \begin{pmatrix} a_{1}\\a_{2}\\a_{3} \end{pmatrix} × \begin{pmatrix} b_{1}+c_{1} \\...
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### Is it necessary to show 0 is included in a set to show it is a vector subspace?

To show a set is a vector subspace, I see it´s necessary to prove a) An addition property: if $x$ and $m$ are both elements of the set, then $x+m$ must also be an element of the set for it to be a ...
1 vote