Search Results

Let $G,H$ be groups and suppose that $\varphi: G\to H$ is a bijection such that for any proper subgroup $G'\neq G$ of $G$ the image $\varphi(G')$ is a subgroup of $H$ and the restriction $\varphi|_{G' …

Let $(\mathbb{Z}/n\mathbb{Z})^*$ denote the multiplicative group of units of $\mathbb{Z}/n\mathbb{Z}$. Is there a finite commutative group $G$ such that for all $n\geq 2$, there is no injective group …

Let $\{0,1\}^{<\omega}$ denote the subgroup of $\{0,1\}^{\omega}$ where $f:\omega\to\{0,1\}$ is a member of $\{0,1\}^{<\omega}$ if there is $N\in\omega$ such that $f(n)= 0$ for all $n\in\omega$ with …

Is it true that there exist $2^{\aleph_0}$ pairwise non-isomorphic torsion-free countable groups?

Are there (infinite) non-isomorphic groups $G, H$ such that there are surjective group homomorphisms $f: G\to H$ and $g: H\to G$?

Which integers $n>2$ have the following property? There is a group $G$ such that $G^n \cong G$; and for all integers $k$ with $1<k<n$ we have $G^k\not \cong G$.

The starting point of this question is the following: If $G$ is a group such that all elements have order at most $2$, then $G$ is commutative. If $G$ is any group, let $G_{>2}$ denote the set o …

We say that a permutation $\varphi:\mathbb{N}\to\mathbb{N}$ is finitary if there is $k\in\mathbb{N}$ such that $\varphi(i) = i$ for all $i\in\mathbb{N}$ with $i\geq k$. Let $I_\mathbb{N}$ denote the g …

Given a non-trivial group $G$ and $g\in G\setminus \{e_G\}$ where $e_G$ is the neutral element, it is easy to show using Zorn's Lemma, that there is a subgroup not containing $g$ that is maximal among …

Is there an infinite non-commutative group $G$ such that every proper subgroup of $G$ is commutative?

Let $X$ be a set and let $(X^X,\circ)$ denote the monoid of all maps $f: X\to X$, together with composition. Let $(\text{Sym}(X),\circ)$ be the group of all bijections from $X$ to itself. Does there …

The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper subgr …

Given a set $X$ is it provable in $\mathsf{ZF}$ that there is a binary operation $\ast: X\times X\to X$ such that $(X,\ast)$ is a group?

Fix a cardinal $\kappa$ and consider $\mathbb{Z}^\kappa$ with componentwise addition and the subgroup $$F_\kappa :=\{g:\kappa \to \mathbb{Z}: \{\alpha\in \kappa: g(\alpha)\neq 0\} \text{ is finite}\}. …

For any non-empty set $X$ let $\text{Sym}(X)$ denote the group of bijections $f:X\to X$ with composition. Is there an infinite set $X$ and a surjective group homomorphism $\pi: \text{Sym}(X)\to \math …