# Tagged Questions

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### What would be an example such that $aH=bH$ but $Ha \neq Hb$?

Let $H$ be a subgroup of a group $G$. Assume $aH=bH$ for some $a,b\in G$ What would be an example such that $Hb\neq Ha$? I cannot imagine what would be.. Moreover, if $H$ is infinite, ...
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### Inner Product on a Vector Space over a field besides $\mathbb R$ or $\mathbb C$?

Are there any fields with vector spaces you can define an inner product over besides subfields of $\mathbb C$? I know that you'd want the field to contain an ordered subfield, so it must have ...
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### If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
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### A question about polynomials in $K[x_1,x_2,…,x_n]$ and there permutations

Let $K$ be a field. Let $n$ be a positive integer and $P$ be a non-symmetric polynomial in $K[x_1,x_2,...,x_n]$. $S_n$ acts on $K[x_1,...,x_n]$ in an obvious way. Let $P_1,P_2,...,P_r$ be the ...
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### Heineken and Mohamed Groups

I was trying to understand the costruction of the Heineken and Mohamed groups. (II) Every proper subgroup of $G$ is subnormal and nilpotent. Lemma 1. If $G$ is a non-nilpotent soluble group ...
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### Can we make $\mathbb{Z}$ into a field?

This is probably an elementary question about fields, but I think it is a little tricky. Can we make the integers $\mathbb{Z}$ into a field? Let me be more precise. Is it possible to make ...
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Is there a nice example of an additive category $C$ and a family of morphisms $S\subset Mor(C)$ such that $C[S^{-1}]$ is no longer additive? I know that in general localization of categories ...
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### Weakened associativity axiom: $(x * y) * z \leq x*(y*z).$

Call a partially ordered magma $(X,*)$ sub-associative iff it satisfies the following axiom. $$(x*y)*z \leq x*(y*z).$$ Basically, this is saying that we may shuffle brackets right to get a larger ...
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### Counterexamples to Nakayama's Lemma if $M$ is not finitely generated

One of the most famous forms of Nakayama's lemma says: Let $I$ be an ideal in $R$ and $M$ a finitely-generated $R$ module. If $IM = M$, then there exists an $r \in R$ with $r ≡ 1 \pmod I$, ...
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### Counterexample to linear transformation.

What counterexample can I use to prove that ($\mathbb{R}_{[x]}$is any polynomial): $L :\mathbb{R}_{[x]}\rightarrow\mathbb{R}_{[x]},(L(p))(x)=p(x)p'(x)$ is not linear transformation. I have already ...
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### If all the centralizers are finite, then does it follow that the group is finite?

As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. ...
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### What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, ...
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### What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$? [closed]

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$ ? Im sorry I am confused about ring theory. Its all new to me.
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### Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an ...
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### Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
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### Worst category with first isomorphism?

I am no expert in category theory, but from VIII of Algebra: Chapter 0 I learnt that In an abelian category every $A\xrightarrow{\phi}B$ can be decomposed into A\twoheadrightarrow ...
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### Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
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### Other Euler characteristics?

At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, ...
I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies Let $N\triangleleft G$. Then ...
### Duality between $[G,G]$ and $Z(G)$? [duplicate]
Possible Duplicate: Center-commutator duality Let $G$ be a group. It seems that there is a certain duality between two of its normal subgroups, the commutator ...