hho that 1,3, 4,5,9} is a group under multiplioation modulo 11

To prove that a set is a group under a certain operation, it must satisfy four properties: closure, associativity, identity, and invertibility. Let's check each of these for the set {1,3,4,5,9} under multiplication modulo 11.

1. Closure: For every pair of elements in the set, their product modulo 11 should also be in the set. Let's check a few examples:

   • 1*3 = 3 (mod 11), which is in the set.
   • 4*5 = 20 = 9 (mod 11), which is in the set.
   • 9*3 = 27 = 5 (mod 11), which is in the set.

   You can check the rest, but it seems like this property is satisfied.

2. Associativity: This is a property of the operation, not the set. Multiplication is associative, so this property is satisfied.

3. Identity: There should be an element in the set such that multiplying any element by the identity gives the same element. In this case, 1 is the identity, because 1*a = a (mod 11) for any a in the set.

4. Invertibility: For every element in the set, there should be another element such that their product is the identity. Let's check:

   • The inverse of 1 is 1, because 1*1 = 1 (mod 11).
   • The inverse of 3 is 4, because 3*4 = 12 = 1 (mod 11).
   • The inverse of 4 is 3, because 4*3 = 12 = 1 (mod 11).
   • The inverse of 5 is 9, because 5*9 = 45 = 1 (mod 11).
   • The inverse of 9 is 5, because 9*5 = 45 = 1 (mod 11).

So, it seems like the set {1,3,4,5,9} is indeed a group under multiplication modulo 11.