### On special subgroups of fundamental group

#### Abstract

Suppose $\alpha$ is a nonzero cardinal number,

$\mathcal I$ is an ideal on

arc connected topological space $X$, and

${\mathfrak P}_{\mathcal I}^\alpha(X)$ is the subgroup of $\pi_1(X)$

(the first fundamental group of $X$) generated by homotopy classes of

$\alpha\frac{\mathcal I}{}$loops.

The main aim of this text is to study ${\mathfrak P}_{\mathcal I}^\alpha(X)$s

and compare them.

Most interest is in $\alpha\in\{\omega,c\}$ and $\mathcal

I\in\{\mathcal P_{fin}(X),\{\varnothing\}\}$, where $\mathcal

P_{fin}(X)$ denotes the collection of all finite subsets of $X$.

We denote ${\mathfrak P}_{\{\varnothing\}}^\alpha(X)$ with

${\mathfrak P}^\alpha(X)$. We

prove the following statements:

\\

$\bullet$ for arc connected topological spaces $X$ and $Y$

if

${\mathfrak P}^\alpha(X)$ is isomorphic to ${\mathfrak P}^\alpha(Y)$

for all infinite cardinal number $\alpha$, then

$\pi_1(X)$ is isomorphic to $\pi_1(Y)$;

\\

$\bullet$ there are arc connected topological spaces $X$ and $Y$

such that $\pi_1(X)$ is isomorphic to $\pi_1(Y)$ but

${\mathfrak P}^\omega(X)$ is not isomorphic to ${\mathfrak P}^\omega(Y)$;

\\

$\bullet$ for arc connected topological space $X$ we have

${\mathfrak P}^\omega(X)\subseteq{\mathfrak P}^c(X)

\subseteq\pi_1(X)$;

\\

$\bullet$ for Hawaiian earring $\mathcal X$, the sets

${\mathfrak P}^\omega({\mathcal X})$, ${\mathfrak P}^c({\mathcal X})$,

and $\pi_1({\mathcal X})$

are pairwise distinct.

\\

So ${\mathfrak P}^\alpha(X)$s and ${\mathfrak P}_{\mathcal I}^\alpha(X)$s

will help us to classify the class of all arc connected topological spaces with

isomorphic fundamental groups.

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