Prove that if in a certain set with associative operation $*$ defined, $ax = b$ and $ya = b$ have solutions, then that structure is a group.
102859
Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions.
27248
If $A$ is an $m \times n$ matrix, prove that $Ax=0$ has infinite solutions $\iff$ $\text{rank}(A)<n$
146231
Let X be a set with an associative binary operation ◦ with identity e. Suppose every element of X satisﬁes x ◦ x = e. Prove that ◦ is commutative.
90434
Prove for every two sets $A$ and $B$ that $A-B$, ${B-A}$ and $A \cap B$ are pairwise disjoint.
36117
Prove that matrices of the form $\begin{pmatrix} x & x \\ x & x \end{pmatrix}$ are a group under matrix multiplication.
68732
Prove that a semigroup satisfying $a^pb^q=ba$ is commutative
49578
Proving $a\mathbb{Z}\cap b\mathbb{Z}=[a,b]\mathbb{Z}$
44258
Conditions under which group solvability is invariant under an homomorphism.
39746
How to prove that given set is a connected subset of the space of matrices?
43759
Prove that the set $\mathrm{Aut}(G)$ of all automorphisms of the group $G$ with the operation of taking the composition is a group
150738
Proof of $(A - B) - C = A - (B \cup C)$
144266
set theory proof of $A\cap B = A \setminus(A\setminus B)$
102383
Associativity of Cartesian Product
117701
Let $G$ be a group and $X$ set. $a,b \in X$ are in the same orbit, so show that $stab(a) \cong stab (b)$
109745
If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
44405
Prove XOR is commutative and associative?
75117
$\mathrm{rank}(AB-BA)=1$ implies $A$ and $B$ are simultaneously triangularisable