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Abstract Algebra

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Moderate

Middle Cancellation Law for Groups

ABSALG-@CU3OP

Which of the following is true?

A

There is a group $G$ such that:

$a\circ x\circ b=c\circ x\circ d$ implies $a\circ b=c\circ d$

does not hold for any $x$ in $G$.

B

There is a group $G$ such that:

$a\circ x\circ b=c\circ x\circ d$ implies $b=d$

does not hold for any $a$, $c$, $x$ in $G$.

C

There is a group $G$ such that

$a\circ x\circ b=c\circ x\circ d$ implies $a=c$

does not hold for any $b$, $d$, $x$ in $G$

D

$G$ is a group such that

for all $a, b, c, d, x\in S$ we have $a\circ x\circ b=c\circ x\circ d$ implies $a\circ b=c\circ d$

if and only if $G$ is an abelian group, meaning $g\circ h=h\circ g$ for all $g, h\in G$.