Mathematics Problem Archive

As in 17.57, let $r(m)=\{r+km\mid k\in\mathbb Z\}$ for integers $0\leqslant r<m$; for $r_1(m_1)\cap r_2(m_2)=\emptyset$ let the class transposition $\...

Suppose that $U$ is a nonprincipal ultrafilter on $\omega$, and $B$ is a group such that every element $b\in B$ belongs to a subgroup of $B$ that is a...

Suppose $\alpha$ is an endomorphism of a group G such that for every group H and every homomorphism $f:G\to H$, there exists an endomorphism $\beta_f$...

Suppose B is a subgroup of the symmetric group $S_\Omega$ on an infinite set $\Omega$. Will the amalgamated free product $S_\Omega *_B S_\Omega$ of tw...

Suppose that G is a finite group, and $A_1,A_2,A_3$ are subsets of G such that the multiplication map $A_1\times A_2\times A_3\to G$ is bijective. Mus...

For a finite group G, the power graph P(G) is the graph with vertex set G and edges \{x, y\} for all $x\ne y\in G$ such that either $x\in\langle y\ran...

Conjecture: Suppose that for a fixed positive integer $k$ at least half of the elements of a finite group $G$ have order $k$. Then $G$ is solvable....

We say that a product $XY=\{xy\mid x\in X,\ y\in Y\}$ of two subsets $X,Y$ of a group $G$ is direct if for every $z\in XY$ there are unique $x\in X$, ...

Let $G$ be an extension of a normal elementary abelian subgroup $A$ by an elementary abelian group $B\cong G/A$ such that $A$ contains an element $a$ ...