Question

Show that {1, 2, 3} under multiplication modulo 4 is not a group
but that {1, 2, 3, 4} under multiplication modulo 5 is a group.​

Answer 1

Answer:

see below

Step-by-step explanation:

since 2 X 2 ≈ 0 ( mod 4 )

and 0 is not in {1, 2, 3} hence not closed.

therefore....

{1, 2, 3} is not a group

but 5 is prime number and mod5 leaves all remainders b/w 1 to 4 only. it also obeys other 3 conditions also

hence

{1, 2, 3, 4} forms a group

Answer 2

Answer:

$[1,2,3]$ is not a group and $[1,2,3,4]$ forms a group were shown.

Step-by-step explanation:

Since $2$ × $2$ ≅ $0$ ( mod $4$ )

and $0$ is not in $[1,2,3]$ hence not closed.

Therefore, $[1,2,3]$ is not a group.

But $5$ is a prime number and mod $5$ leaves all remainders bertween $1$ to $4$ only.  It also obeys other $3$ conditions also.

Hence, $[1,2,3,4]$ forms a group.