Proving a = a^-1 implies a^2 = e in group theory

Thread starter mehrts
Start date Nov 16, 2006
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Group Group theory Theory

In summary, Group theory is a branch of mathematics that studies the properties of groups, which are mathematical objects that consist of a set of elements and a binary operation. A simple group is a group that does not have any nontrivial normal subgroups and cannot be broken down into smaller groups. To determine if a group is simple, one needs to check for nontrivial normal subgroups. Group theory has applications in various fields such as physics, chemistry, computer science, and cryptography. An example of a simple group is the alternating group, A<sub>n</sub>, which consists of even permutations of n objects and is simple for n ≥ 5.

Nov 16, 2006

mehrts

Let G be a group. Let 'a' be an element of G. Let e be the identity of G Prove that if a = a^-1 then a^2= e.

Is the proof below correct ?
Suppose a = a^-1. Then

a^2 = aa = a(a^-1) = e.

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Nov 16, 2006

buzzmath

yeah, if a is it's own inverse then a^2 is e. your proof looks fine to me.

Nov 16, 2006

mehrts

Thxs. He he he :!)

Related to Proving a = a^-1 implies a^2 = e in group theory

1. What is group theory?

Group theory is a branch of mathematics that studies the properties of groups, which are mathematical objects that consist of a set of elements and a binary operation (such as addition or multiplication) that satisfies certain axioms.

2. What is a simple group?

A simple group is a group that does not have any nontrivial normal subgroups, meaning that there are no proper subgroups that are invariant under the group's operation. In other words, a simple group cannot be broken down into smaller groups.

3. How do you determine if a group is simple?

To determine if a group is simple, you need to check if there are any nontrivial normal subgroups. This can be done by examining the elements of the group and seeing if there are any subgroups that are invariant under the group's operation. If no such subgroups exist, then the group is simple.

4. What are the applications of group theory?

Group theory has many applications in various fields, including physics, chemistry, computer science, and cryptography. It is used to study the symmetry of physical systems, analyze molecular structures, and develop algorithms for data encryption.

5. Can you give an example of a simple group?

One example of a simple group is the alternating group, denoted as A_n, which consists of all even permutations of n objects. This group is simple for n ≥ 5.