Suppose $G$ is a group with the following properties. $G/Z(G)$ is a Tarski $p$-group or another simple finitely generated infinite group in which all proper subgroups are abelian, and $Z(G)$ is a direct sum of two cyclic groups $\langle c_1\rangle$ and $\langle c_2\rangle$ of order $p$, a prime.

Clearly setting $c_1\mapsto c_1c_2$ and $c_2\mapsto c_2$ will give us an automorphism of $Z(G)$. Is it possible, in these or similar circumstances to extend this automorphism to an automorphism of the whole group $G$?

Are there some sufficient conditions to do it, or one can only hope for some magic?

If $P$ is a polynomial over the field $\mathbb F_q$ of degree at most $q-2$ with $k$ nonzero coefficients, then $P$ has at most $(1-1/k)(q-1)$ distinct nonzero roots.

Does this fact have any standard name / reference / proof / refinements / extensions?

I am interested the following element of the group algebra $\mathbb{Q}S_n$: \begin{align} \phi_n=2e+(12)+(123)+\dots+(1\dots n) \end{align} where $e$ is the identity permutation. My question is whether $\phi_n$ is a unit.

For small $n$ I can see numerically that $\phi_n$ is a unit, but I have no idea how to prove it for general $n$. As a linear map from $\mathbb{Q}S_n$ to itself, for $n<10$, $\phi_n$ seems to be diagonalisable with positive integer eigenvalues, and I have no idea why this should be.

Consider the function

$f_k(c):=\sum_{n=0}^{\infty} c^{n^k}$ where $k\ge 1$ is an integer. This one obviously converges for $\left\lvert c \right\rvert <1.$

In the following we want to study the solution to the equation

$$\sum_{n=0}^{\infty} \left(n^k -\frac{1}{\alpha}\right) c^{n^k}=0.$$

This one always exists as long as $\alpha \in (0,1).$

Numerically, I discovered something that I would like to understand:

As $\alpha \rightarrow 0$ we have that $c= 1-1/k \gamma \alpha.$

So first the solution $c$ seems to depend in a linear way on $\alpha$ for $\alpha$ small and second, the dependence on $k$ also seems to be just $1/k$.

I would like to understand these two observations.

Of course, I have studied this more carefully, but just as a quick sanity check, one can study in Mathematica

$$\sum_{n=0}^{1000} \left(n^k -\frac{1}{0.01}\right) c^{n^k}=0.$$

For **k=1**:

FindRoot[!( *UnderoverscriptBox[([Sum]), (i = 0), (1200)](c^i\ ((i - *FractionBox[(1), (a)])))) == 0, {c, 1}]

this produces **c=0.9901**,

For **k=2**:

*FindRoot[!(
*UnderoverscriptBox[([Sum]), (i =
0), (1200)](c^((i^2))\ ((i^2 -
*FractionBox[(1), (a)])))) == 0, {c, 1}]*

this produces **c=0.9952**,

and for **k=4**:

*FindRoot[!(
*UnderoverscriptBox[([Sum]), (i =
0), (1200)](c^((i^4))*\ ((i^4 -
*FractionBox[(1), (a)])))) == 0, {c, 1}]*

this produces: **c=0.99777**

Does anybody have any insights on this?

Recall a very famous theorem due to Hartogs for complex analytic functions of several variables.

**Hartogs's Theorem** Let $f$ be a holomorphic function on a set $G \setminus K$, where $G$ is an open subset of $\mathbb{C}^n$ ($n \ge
2$) and $K$ is a compact subset of $G$. If the complement $G\setminus
K$ is connected, then $f$ can be extended to a unique to a unique
holomorphic function on $G$.

My question is does this theorem still hold if $f$ real-analytic?

By real-analytic, I mean that $f$ has a power series expansion on $G\setminus K$.

Let $\mathscr{C}$ be a category and $(W,C,F)=$(weak equivalence,cofibration,fibration) is some model structure of $\mathscr{C}$. Another model structure of $\mathscr{C}$, $(W_{S},C,F_{S})$ are called stronger if $W \subset W_{S}$. $Ex$ and $Ex_{S}$ are fibrant replacement functors in each model structures. When $X\longrightarrow Y$ belongs to $W_{S}$, $Ex_{S}(X)\longrightarrow Ex_{S}(Y)$ belong to $W$. Since $F=RLP(C\cap W)$ and $W \subset W_{S}$, $F_{S}\subset F$.

In my situation, $\mathscr{C}$ is the category of pointed or unpointed simplicial sets, $(W,C,F)$ is ordinary model structure and $(W_{S},C,F_{S})$ is its localization by some monomorphism $f\colon A\longrightarrow B$ and $W_{S}=\{f$-local equivalence} and $F_{S}=RLP(C\cap W_{S})$.

Q. Is $Ex_{S}(F) \subset F_{S}$ ? In other words, when $X\longrightarrow Y$ is a fibration, is $Ex_{S}(X)\longrightarrow Ex_{S}(Y)$ a stronger fibration?

I have the following question the answer to which I cannot find in the literature (but it must have been studied):

Suppose that $M\subset\mathbb{C}^2$ is a real surface which may locally be written as the zero set of a holomorphic function. What conditions are required on $M$ so that $M$ may be written as the restriction of a projective algebraic curve $C\subset\mathbb{CP}^2$ to $\mathbb{C}^2\subset\mathbb{CP}^2$ (where we identify $\mathbb{C}^2$ with $\{[z,w,1]\in \mathbb{CP}^2| z,w \in \mathbb{C}\}$)?

Clearly some conditions are necessary for example by considering $M=\{(z,e^z)\in \mathbb{C}^2|z\in\mathbf{C}\}$. I expect something like finite Euler characteristic, finite density at infinity, is this enough? Is there some standard reference for this? Or is there some sneaky way to use the Chow theorem?