$23.$ The following $$definitions$$ of a group are taken verbatim, including spelling and punctuation, from papers of

students who wrote a bit too quickly and carelessly. Criticize them.

a$.$ A group G is a set of elements together with a binary operation $$ such that the following conditions are

satisfied

$$ is associative

There exists e in G such that

e $$ x $=$ x $$ e $=$ x $=$ identity.

For every a in G there exists an a $($inverse$)$ such that

a $$ a $=$ a $$ a $=$ e

b$.$ A group is a set G such that

The operation on G is associative.

there is an identity element $($e$)$ in G$.$

for every a in G$,$ there is an a $($inverse for each element$)$

c$.$ A group is a set with a binary operation such

the binary operation is defined

an inverse exists

an identity element exists

d$.$ A set G is called a group over the binery operation $$ such that for all a$,$ b in G

Binary operation $$ is associative under addition

there exist an element $\{$e$\}$ such that

a $$ e $=$ e $$ a $=$ e

Fore every element a there exists an element a such that

a $$ a $=$ a $$ a $=$ e