- #1
HuaYongLi
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Homework Statement
G is a commutative group, prove that the elements of order 2 and the identity element e form a subgroup.
Homework Equations
The Attempt at a Solution
I don't know where to even begin.
A subgroup is a subset of a group that has the same group operation as the larger group and contains the identity element. It is a smaller group within a larger group.
To prove that a set of elements is a subgroup of a group, you need to show that it satisfies the three conditions of a subgroup: closure, associativity, and existence of an identity element. You also need to show that every element in the subset has an inverse in the subset.
The order of an element in a group is the smallest positive integer n such that the element raised to the power of n is equal to the identity element. In simpler terms, it is the number of times you need to combine the element with itself to get the identity element.
To prove that a subgroup only contains elements of order 2 and the identity element, you need to show that every element in the subset has an order of 2 or is the identity element. This can be done by raising each element to the power of 2 and showing that it equals the identity element.
No, a subgroup can only have elements of the same order or the identity element. This is because a subgroup must contain the identity element and the inverse of every element in the subset, which both have an order of 1. Therefore, all elements in the subset must have the same order.