For G = <a: a^100 = e>, find the following

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In summary, the notation "G = <a: a^100 = e>" represents a finite group where the identity element is e. In this context, "find the following" means to determine the value or element that satisfies the given condition. To solve a problem involving finding an element in a group, you can use the properties of the group and the given information to manipulate the equation. One example of a group where the equation holds true is the group of integers under addition modulo 100. There are faster ways to find the solution to this problem by utilizing properties and theorems related to groups.
  • #1
CoachBryan
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It's asking me to find a^-5
From what I understood about inverses is that it's the element before you hit the identity would be the inverse. So a^5 = (a^5, a^10, a^15,..., a^90, a^95, a^100 = e), then a^-5 = a^95.

The correct answer is actually a^20.

Can anyone help me out with understanding inverses? Thanks!
 
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  • #2
Nevermind, it was a typo. The answer was a^-5 = a^95
 

FAQ: For G = <a: a^100 = e>, find the following

What is the significance of the notation "G = "?

The notation "G = " represents a group, where G is the name of the group and the elements of the group are represented by the symbol "a". The equation "a^100 = e" indicates that the group is finite and the identity element is e.

What does the term "find the following" mean in this context?

In this context, "find the following" means to determine the value or element that satisfies the given condition. In this case, we are looking for the element a that, when raised to the 100th power, is equal to the identity element e.

How do you solve a problem involving finding an element in a group?

To solve a problem involving finding an element in a group, you can use the properties of the group and the given information to manipulate the equation and find the value of the unknown element.

Can you provide an example of a group where the given equation holds true?

One example of a group where the equation "a^100 = e" holds true is the group of integers under addition modulo 100. In this group, the identity element is 0 and any integer raised to the 100th power modulo 100 will result in 0.

Is there a faster way to find the solution to this problem?

Yes, there are certain properties and theorems related to groups that can help you find the solution more efficiently. For example, the order of an element in a group must divide the order of the group. So in this case, since the order of the group is 100, the order of element a must be a factor of 100, making it easier to narrow down the possible values of a.

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