halvizo1031 said:Homework Statement
suppose that G is a group in which every non-identity element has order two. show that G is commutative.
Homework Equations
The Attempt at a Solution
IS THIS CORRECT?
ab=a[(ab)^2]b=(a^2)(ba)(b^2)=ba
The order of an element in a group refers to the number of times that element must be combined with itself in order to reach the identity element. In this case, "order 2" means that the element combined with itself only twice will result in the identity element.
Yes, a group can have multiple elements with order 2. For example, the group of integers under addition has two elements with order 2: 1 and -1.
Groups with elements of order 2 can be represented using a Cayley table, which is a visual representation of the group's operation table. Alternatively, they can also be represented using permutation notation or through matrix representations.
No, the identity element is not always the only element with order 2 in a group. In fact, in groups with an even number of elements, there will always be at least one other element with order 2.
Groups with elements of order 2 have various applications in fields such as computer science, cryptography, and physics. For example, they are used in encryption algorithms and in quantum computing to represent quantum states.