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Bds_Css
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Homework Statement
Prove that a non-abelian group of order 10 must have an element of order 2.
What if the order of every element is 5?
Prove there are 5 elements of order 2.
Bds_Css said:What if the order of every element is 5?
Prove there are 5 elements of order 2.
Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It deals with the study of mathematical objects and their properties, rather than specific numbers or equations.
A non-abelian group is a mathematical structure consisting of a set of elements and an operation that is not commutative. This means that the order in which the elements are combined affects the outcome of the operation.
The order of an element in a group is the smallest positive integer n such that the element raised to the power of n equals the identity element of the group. In other words, if an element has order 2, it means that when it is multiplied by itself, the result is the identity element.
This is a theorem in abstract algebra known as the Cauchy's theorem. It states that if a prime number divides the order of a group, then the group must contain an element of that prime order. In this case, since 2 is a prime number and it divides the order of the group, the group must contain an element of order 2.
One example of such a group is the dihedral group D5, which consists of the symmetries of a regular pentagon. This group has 10 elements and one of them, the rotation by 180 degrees, has order 2.