Inverses of elements of a group

In summary, the conversation is about finding the inverse for each element in the group {1, 9, 16, 22, 29, 53, 74, 79, 81} under modulo 91. The equations used to find the inverse are discussed, with the requirement that an element must have a greatest common divisor of 1 with 91 to be invertible. The process of finding the inverse is also explained, with an example using the elements 9 and 81. Finally, the individual asks for help on another post related to group theory.
  • #1
duki
264
0

Homework Statement



Find the inverse for each element in the group {1, 9, 16, 22, 29, 53, 74, 79, 81}, which is under modulo91

Homework Equations



The Attempt at a Solution



From notes (again):
1 = 1
9 = 81
16 = 74
22 = 29
53 = 79

How were these numbers found? Sorry for so many questions, I'm just really lost on this group stuff. Thanks :)
 
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  • #2
Well, again, an element of V_91, call it a, is invertible if and only if gcd(a,91)=1.

Find such elements first.
 
  • #3
Does that hold true for all the elements? All of them have gcd(a,91) = 1
 
  • #4
I ndeed to correct myself on my previous post. It should have read: an element of Z_91, call it a, is invertible iff gcd(a,91)=1. Because it doesn't make sense to say an elment of V_91, since we know that V_91 is the set of all invertibles of Z_91.

Now the task of findiing their inverses is another issue. What you need to do is find two elements a,b of V_91 such that

[a]=[1] that is, two elements such that when you multiply them and then take mod 91 they should give u 1.

Say 9 and 81=> 9*81=729=> 729=8*91+1=> so the remainder is 1, which means that

729=1(mod 91) so 9, and 81 are multiplicative inverses of each other.
 
  • #5
Excellent, thanks! That makes perfect sense now.

If it's not too much trouble, there's another (3rd) post I made that has worked its way down the board, also on group theory. Could you maybe give me a hand on that one?
 

1. What is an inverse element of a group?

An inverse element of a group is an element that, when combined with another element in the group using the group's operation, results in the identity element. In other words, it "undoes" the action of the original element.

2. How do you find the inverse of an element in a group?

To find the inverse of an element in a group, you can use the group's operation to combine the element with itself until you reach the identity element. The resulting element will be the inverse.

3. Can every element in a group have an inverse?

Yes, every element in a group has an inverse. This is because a group is defined as a set of elements with a binary operation that satisfies the group axioms, one of which states that every element must have an inverse.

4. What is the notation for an inverse element in a group?

The notation for an inverse element in a group is to include a superscript -1 after the element. For example, if the element is denoted as "a", its inverse would be denoted as "a-1".

5. Can an element in a group have more than one inverse?

No, an element in a group can only have one inverse. This is because an inverse element must result in the identity element when combined with the original element, and the identity element can only have one result when combined with any element in the group.

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