Linked Questions

Let $G,H$ groups. Suppose that $G \times G \cong H \times H$. Then is necessarily $G \cong H$? I know that if $G,H$ are finite groups then it is true (you show that the map $FinGrp \rightarrow \...

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 $G^k\not \cong G$.

Let $G$ be a group. Suppose that $G\simeq G\times G\times G$ (here $\simeq$ is an isomorphism of groups). Is it true that in this case $G\simeq G\times G$? Of course, this question is slightly ...

There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?

(This is a re-post of my old unanswered question from Math.SE) For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types. Recall that: $0,...

What's the simplest example (if any) of two non-isomorphic groups G and H such that $G \times G \cong H \times H$? A similar question can be asked for $n^{th}$ powers for fixed $n > 1$. The Krull-...

Which cardinals $\lambda > 2$ have the following property? There is a space $(X,\tau)$ such that for all cardinals $\kappa$ with $1<\kappa<\lambda$ we have $X\not\cong X^\kappa$, and $X\cong ...

Suppose $M$ is a finitely generated left module over a ring $R.$ We define the rank of $M$ as the minimal number of generators of $M.$ If in addition $M$ is free, then we define the free-rank of $...

Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...

We call an topological space $(X,\tau)$ $n$-product-periodic for an integer $n\geq 3$ if $\prod_{i=1}^n X \cong X$ but for all integers $k$ with $2\leq k\leq n-1$ we have $\prod_{i=1}^k X \not\cong X$....

Gowers and Maurey proved in their remarkable paper(s), that there is a Banach space $X$ such that $X$ is isomorphic to its cube $X\oplus X\oplus X$ but not to isomorphic to its square $X\oplus X$. ...

Possible Duplicate: when is A isomorphic to A^3? Does there exist a group $G$ such that $G \cong G \times G \times G$ and $G \not \cong G \times G$? If such groups exist, can $G$ be countable? ...

According to answers to this Math Overflow question, there is an infinite rank abelian group $A$ such that $A\cong A^3$ but $A\not\cong A^2.$ Clearly $A$ is an retract of $A^2$ while $A^2$ is an ...

A group $H$ is called a retract of a group $G$ if there exist homomorphisms $f:H\longrightarrow G$ and $g:G\longrightarrow H$ such that $g\circ f=id_H$. By a trivial retract of $G$, I just mean the ...