3
votes
0answers
29 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
3
votes
0answers
38 views

Existence of a natural transformation

Let $G$ be a group, $S$ a set and $\chi$ an action of $G$ on $S$. This defines a functor $F$ from the group as a single object category to the category of sets. Let $\phi$ be an automorphism of $G$, ...
3
votes
1answer
39 views

Pull-backs of diagrams of groups with free product.

Until recently I calculated only pull-back of diagrams of finite groups. Now I am trying to calculate the pull-back of diagram of groups when the groups are free products of other groups. It seems ...
2
votes
0answers
44 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
2
votes
0answers
36 views

Push-out of product of diagrams

Im working on the category of groups. Let $D$ be the push-out of the diagram $B\leftarrow A \rightarrow C$. Let $D'$ be the push-out of the diagram $B' \leftarrow A' \rightarrow C'$. It is possible to ...
1
vote
1answer
38 views

Explanation of a natural transformation in group theoretic terms

In category theory, a functor looks like homomorphism between categories. Keeping that analogy in mind, can a natural transformation be described by (or restricted to) group theoretic terms? For ...
3
votes
1answer
35 views

Terminology concerning conjugation in groups of functions.

If there is a function $a$ such that $a\circ g\circ a^{-1}=h$ then the functions $g$ and $h$ are conjugate to each other. If one wished to identify $a$, would one say "$g$ and $h$ are conjugate "by ...
2
votes
1answer
58 views

is the abelianization functor (on groups) full?

By abelianization I mean, for any group $G$, its commutator subgroup is the subgroup $[G,G]$ generated by elements of the form $ghg^{-1}h^{-1}$ for $g,h\in G$. Then the abelianization of $G$ is ...
2
votes
1answer
49 views

Group categories with only one object with a defined product

Do you know how to deal with this kind of problem? Let $G$ be a group and $\mathcal {G} $ be the category with one object $G$ and $\mbox {Mor}( {G} ; {G} ) = {G} $. Find all groups $G$ such that ...
2
votes
0answers
22 views

Unipotent Group Scheme

Can someone give me a reference of why this is true? Let A an associative k-algebra in wich every element is nilpotent, (non unital) and free of finite rank k-module, then for every commutative ...
5
votes
1answer
83 views

Nonexistence of a functor from $Group$ to $Set$ taking each group to its set of automorphism

I am struggling with this question: show that there does not exist functor from $Group$ to $Set$ taking each group to its set of automorphisms. I have thought about it for a while now, not having any ...
1
vote
0answers
31 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
12
votes
5answers
635 views

Abelianization of free group is the free abelian group

How does one prove that if $X$ is a set, then the abelianization of the free group $FX$ on $X$ is the free abelian group on $X$?
10
votes
1answer
98 views

Field Extensions and their dimensions

There is a powerful theorem with respect to a field $F$ extensions and their dimensions. $F<E<K \ \Rightarrow [K:F] = [K:E][E:F] $ This is analogous to the famous Lagrange's theorem with ...
2
votes
1answer
115 views

Free product of groups as coproduct

Wikipedia says "The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusion maps. Unlike direct products, coproducts in other categories are not all ...
10
votes
0answers
122 views

Cogroup structures on the profinite completion of the integers

Let $\mathsf{ProFinGrp}$ be the category of profinite groups (with continuous homomorphisms). This is equivalent to the Pro-category of $\mathsf{FinGrp}$. Notice that $\widehat{\mathbb{Z}} = ...
10
votes
3answers
200 views

Can we rediscover the category of finite (abelian) groups from its morphisms?

It was a question on stackexchange approximately a month ago if in the category $(grp)^{fin}$ $|Hom(H,G_1)|= |Hom(H,G_2)|$ for all $H \Rightarrow G_1 \cong G_2$. Link to the previous question. So ...
4
votes
1answer
73 views

How general are $\mathrm{Aut}$ groups and $\mathrm{End}$ rings?

For any locally small category $X$ and object $A\in X$ the set $\mathrm{Aut}_X(A)$ is a group w.r.t. composition $\circ$. For any locally small abelian category $X$ and object $A\in X$ the set ...
5
votes
2answers
98 views

Functors and Groups

Let $\alpha$ be a functor from the category of groups in the category of groups which assigns to every group $G$ a characteristic subgroup $\alpha (G)$ of $G$ and to every homomorphism $\theta : H ...
2
votes
1answer
54 views

Group and preorder

Let $G$ be a group. Let $a, b$ be elements of $G$. We denote $\operatorname{Hom}(a, b) = \{ab^{-1}\}$. Then we get a category whose set of objects is $G$. We can regard this category as a preorder in ...
4
votes
2answers
83 views

What can we learn purely from the existence of a (non-constant) functor to the category of abelian groups?

I admit that the following is a very broad question. So if you feel that it is too vague please say so. It might also just be that I haven't read enough about category theory and my question is silly. ...
5
votes
1answer
85 views

A monomorphism of groups which is not universal?

Is there an injective homomorphism of groups $f_1\colon G\longrightarrow H_1$ together with another homomorphism $f_2\colon G\longrightarrow H_2$ such that the pushout $H_2\longrightarrow H_1\coprod_G ...
4
votes
1answer
76 views

Linear structure on the category of formal groups

Let $R$ be a commutative ring. If $R$ is a $\mathbb{Q}$-algebra, then the category of formal groups over $R$ (or the category of formal group laws) carries the structure of an $R$-linear category; ...
9
votes
0answers
127 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
6
votes
2answers
116 views

Computing easy direct limit of groups

How do I start computing easy direct limit of groups: 1) $\mathbb{Z} \overset{1}\longrightarrow \mathbb{Z} \overset{2}\longrightarrow \mathbb{Z} \overset{3}\longrightarrow \mathbb{Z} ...
3
votes
1answer
121 views

Slicker construction of the free product of groups

The usual construction of the free product of groups $\{G_i\}_{i \in I}$ consists of taking the discriminated union $\coprod_{i \in I} G_i$ and taking the set of words satisfying a handful of ...
1
vote
1answer
39 views

Induced injections between free groups

Let $A$ and $B$ be non-empty sets with associated free groups $F(A),F(B)$. Given an injective function $f: A \to B$, is the induced homomorphism $\bar{f}: F(A)\to F(B)$ injective? Let $i_A: A \to ...
7
votes
0answers
164 views

Kernels in $\mathbf{Top}$

There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following : ...
2
votes
1answer
87 views

Understanding Basic Categorical Duality with an Example from Group Theory

I am trying to understand the concept of duality in category theory, but I am having a problem, well illustrated by the following situation. Let $H$ be any nontrivial subgroup of the alternating ...
2
votes
1answer
70 views

pullback square of factor groups

Let H and K be normal subgroup of a group G. The following square is always a pullback square? $$\begin {matrix} G/H\cap K\rightarrow &G/K\\ \downarrow&\downarrow\\ G/H\rightarrow&G/HK\\ ...
5
votes
2answers
69 views

A generalization of abelian categories including Grp

The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp ...
2
votes
1answer
73 views

Proving that tensor distributes over biproduct in an additive monoidal category

I'm trying to prove that the tensor product distributes over biproducts in an additive monoidal category; namely that given objects $A,B,C$, we have $A \otimes (B \oplus C) \cong (A \otimes B) \oplus ...
3
votes
2answers
110 views

Counter examples on Categories

I'm reading Categories for the Working Mathematician by Saunders Mac Lane. At the section 5 from chapter 1, for a fixed category, he claims that every arrow with right inverse, is epic (right ...
5
votes
1answer
129 views

Grothendieck group of a symmetric monoidal category is a lambda ring?

I understand that taking the Grothendieck group of a braided monoidal (abelian) category gives us a commutative ring and that taking that of a symmetric monoidal (abelian) category gives us a ...
9
votes
1answer
183 views

a group is not the union of two proper subgroups - how to internalize this into other categories?

A well-known fact from group theory is that a group cannot be the union of two proper subgroups. I wonder, does this statement internalize into other categories than the category of sets? That is, is ...
3
votes
1answer
127 views

Cocartesian squares in the category of abelian groups.

Recently, I've been doing a recap of (basic) category theory and found an old exercise I seem to be unable to solve. The question is as follows. Let $A, B$ be abelian groups, $A'<A$ and $B'<B$ ...
4
votes
1answer
118 views

Group actions and natural isomorphisms

Let $G$ be a group-as-category, and let $S$ be the image of $G$ by the $Hom(G,-)$ functor. The $Hom$ functor defines a bijection $\chi: G \to S$ between elements of $S$ and morphisms of $G$, and thus ...
5
votes
2answers
113 views

Free objects in $\mathrm{Set}(G).$

What are the free objects in the category of $G$-sets for a group $G$? After considerable deliberation (I'm not very bright), I'm pretty sure they are the $G$-sets $X$ on which $G$ acts freely, that ...
4
votes
2answers
113 views

About the category $\mathrm{Set}(G)$

I'm not good with categories. I've attempted several times to understand what a natural transformation is, and so far I've failed. But I'm trying to learn algebraic topology now, and it seems that I ...
14
votes
0answers
166 views

Existence of a certain functor $F:\mathrm{Grpd}\rightarrow\mathrm{Grp}$

Let $\mathrm{Grpd}$ denote the category of all groupoids. Let $\mathrm{Grp}$ denote the category of all groups. Are there functors $F\colon\mathrm{Grpd}\rightarrow \mathrm{Grp}, ...
4
votes
4answers
98 views

Uniqueness of the operation for a preadditive category?

When working on problems in Rotman's Algebra, he asks us to show that Groups is not a preadditive category. If we could show that the binary operation on $\mathrm{Hom}(A,B)$ had to be $f + g \mapsto ...
2
votes
1answer
89 views

Direct product of groups in categorical terms

In the category of abelian groups $\mathsf{Ab}$ the coproduct is the "direct" product (only finite support), but the product is the "cartesian" product (may be infinite support). In the category of ...
1
vote
3answers
114 views

Finitely generated subgroups of direct limits of groups

Let $G$ be a direct limit of groups $G_n$ for $n\in \mathbb{N}$ (or perhaps even for $n$ in some other directed set, but in my case I only need $\mathbb{N}$). Is it true that every finitely generated ...
1
vote
1answer
44 views

Image of the composition of a kernel with a cokernel.

Let $ h:H\to G $ and $ k:K\to G $ be two normal monomorphisms and let $ f:H\ast K\to G $ theire coproduct. It is always true that $ h\text {coker} k $ and $ f\text {coker} k $ has the same image?
5
votes
1answer
364 views

Equivalences and isomorphisms of short exact sequences

In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
1
vote
1answer
118 views

Direct limit of group

When we study shaves we have that a germ is the direct limit of groups (set, vector spaces). But how can I show that the direct limit of groups is a group?
1
vote
1answer
46 views

Morphisms in the category of group presentations

What are the morphisms in the category of group presentations?
0
votes
1answer
64 views

$M\cong_{R\operatorname{-Mod}}N$ if and only if $M\cong_{\operatorname{Ab}}N$ (Warning: This statement is FALSE!)

I was rather surprised by the fact that two modules are isomorphic if and only if their abelian group structures are isomorphic. I might just sketch the proof here. Given a ring homomorhpism ...
4
votes
1answer
92 views

Quotient group as colimit

I have been wondering for a while about the following question without getting anywhere: Let $G$ be a group, $N$ a normal subgroup. Can the quotient group $G/N$ be seen as the (category theoretical) ...
1
vote
1answer
186 views

Group of Isomorphisms of a Groupoid

Write a careful proof that every group is the group of isomorphisms of a groupoid. In particular, every group is the group of automorphisms of some object in some category. First can someone tell me ...