Intereting Posts

Does a continuous point-wise limit imply uniform convergence?
Asymptotic behaviour of a multiple integral on the unit hypercube
Counting non-isomorphic relations
Probability Theory, Symmetric Difference
Deeply confused about $\sqrt{a^5}=(a^5)^{1/5}$
Proof of Non-Convexity
Indefinite summation of polynomials
Questions on symmetric matrices
How to prove the sum of combination is equal to $2^n – 1$
Heat Equation on Manifold
Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field
Can a number have infinitely many digits before the decimal point?
Functions between topological spaces being continuous at a point?
Bipartite graph non-isomorphic to a subgraph of any k-cube
Why is this polynomial irreducible?

Let $\mathsf{ProFinGrp}$ be the category of profinite groups (with continuous homomorphisms). This is equivalent to the Pro-category of $\mathsf{FinGrp}$. Notice that $\widehat{\mathbb{Z}} = \lim_{n>0} \mathbb{Z}/n\mathbb{Z}$ is a cogroup object, since it represents the forgetful functor $U : \mathsf{ProFinGrp} \to \mathsf{Set}$. Although $\mathsf{ProFinGrp}$ has no coproducts (?), the coproduct $\widehat{\mathbb{Z}} \sqcup \widehat{\mathbb{Z}}$ exists, it coincides by formal nonsense with $\widehat{F_2}$, where $F_2$ is the free group on two generators $x,y$. Let us denote the generator of $\mathbb{Z}=F_1$ by $x$. Then the comultiplication of $\widehat{F_1}$ is given by $\widehat{F_1} \to \widehat{F_2}$, $x \mapsto x * y$. There is another cogroup structure on $\widehat{F_1}$, corresponding to the opposite group. The corresponding comultiplication is $\widehat{F_1} \to \widehat{F_2}$, $x \mapsto y * x$.

**Question.** Are these two the only cogroup structures on $\widehat{\mathbb{Z}}$?

Here is a down-to-earth description what a cogroup structure on $\widehat{\mathbb{Z}}$ is: It is an element $m(x,y) \in \widehat{F_2}$ (a sort of “profinite word” in $x$ and $y$) such that

- Defining Test-Objects
- Example of $A \le G$ solvable, $B \lhd G$ solvable, but $AB$ is not solvable
- Example of infinite field of characteristic $p\neq 0$
- Description of the kernel of the tensor product of two linear maps
- Let $\pi$ denote a prime element in $\mathbb Z, \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
- Isomorphism types of semidirect products $\mathbb Z/n\mathbb Z\rtimes\mathbb Z/2\mathbb Z$

- $m(x,1)=m(1,x)=x$ in $\widehat{F_1}$
- $m(x,m(y,z))=m(m(x,y),z)$ in $\widehat{F_3}$
- There is some $i \in \widehat{F_1}$ such that $m(x,i(x))=m(i(x),x)=1$.

This is very similar to the definition of a one-dimensional formal group law (which is a cogroup structure on $R[[t]]$ in the category of complete topological $R$-algebras).

Background: An affirmative answer would answer math.SE/656279. It is known that $\mathbb{Z} \in \mathsf{Grp}$ has only two cogroup structures.

- Number of irreducible polynomials with degree $6$ in $\mathbb{F}_2$
- Radical extension
- Field reductions
- The total ring of fractions of a reduced Noetherian ring is a direct product of fields
- some question related to $\mathbb{Z}$ and UFD
- Prime ideals in $\mathbb{Z}$
- Category with zero morphisms
- Does a finite ring's additive structure and the structure of its group of units determine its ring structure?
- Does the Dorroh Extension Theorem simplify ring theory to the study of rings with identity?
- Universal Property: do people study terminal objects in $(X\downarrow U)$?

Let $\sigma\in\widehat{\mathbb{Z}}^\times$. It makes sense to raise an element of a profinite group to the $\sigma$ power, and on $\widehat{\mathbb{Z}}$ this is an automorphism. We can build a cogroup structure from the standard one $m(x,y)=xy$ by conjugating by $\sigma$. Explicitly this is $m_{\sigma}(x,y):= (x^{\sigma}y^{\sigma})^{\sigma^{-1}}$. Note that taking $\sigma=-1$ gives your other cogroup law $m_{-1}(x,y)=yx$.

Now $1$ and $-1$ are the only two elements of $\mathbb{Z}^\times$, but $\widehat{\mathbb{Z}}^\times$ has continuum cardinality, so there are lots of other cogroup structures on $\widehat{\mathbb{Z}}$

- compact Hausdorff space and continuity
- Extending a linear map
- How to show that if two integral domains are isomorphic, then their corresponding field of quotients are isomorphic?
- Compare growth rate of functions
- Extending open maps to Stone-Čech compactifications
- How can one change this recurrence relation to be in terms of the first value of a?
- How to endow topology on a finite dimensional topological vector space?
- Showing that $\det(M) = \det(C)$
- Why don't we define division by zero as an arbritrary constant such as $j$?
- Divergence of $ \sum_{n = 2}^{\infty} \frac{1}{n \ln n}$ through the comparison test?
- Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?
- Finding the bound of a linear functional defined on $C$
- Proving whether ideals are prime in $\mathbb{Z}$
- Smallest number $N$, such that $\sqrt{N}-\lfloor\sqrt{N}\rfloor$ has a given continued fraction sequence
- How to define own group action in GAP?