Prove that a group of even order must have an element of order 2.

Prove that a group of even order must have an element of order 2.

Question
Abstract algebra
asked 2021-01-06
Prove that a group of even order must have an element of order 2.

Answers (1)

2021-01-07
To show: thereexist some x in G such that order of x is 2.
Let \(\displaystyle{S}={\left\lbrace{x}\in{G}{\mid}{\left|{x}\right|}{>}{2}\right\rbrace}\)
Let \(\displaystyle{l}\in{S}\)
Since order of any element in group is equal to order of its inverse, thus \(\displaystyle{\left|{l}^{{-{{1}}}}\right|}={\left|{l}\right|}{>}{2}\)
Hence, \(\displaystyle{l}^{{{l}\in{S}}}\)
Since, |l|>2 it implies that \(\displaystyle{l}\ne{l}^{{-{{1}}}}\). Hence every lement in S can be paired off with its inverse which is also on S. Thus, S otains even number of elements
\(\displaystyle\frac{{G}}{{S}}={\left\lbrace{x}\in{G}{\left|{x}={e}{\quad\text{or}\quad}\right|}{x}{\mid}={2}\right\rbrace}\)
Since , identity element is unique in a group, therefore G must have an element of order 2 in order to \(\displaystyle\frac{{G}}{{S}}\) with even number of elements.
Therefore, a group of even order must have an element of order 2.
0

Relevant Questions

asked 2021-03-02

Let g be an element of a group G. If \(|G|\) is finite and even, show that \(g \neq 1\) in G exists such that \(g^2 = 1\)

asked 2020-11-20
Prove that in any group, an element and its inverse have the same order.
asked 2021-01-27
Let G be a group of order \(\displaystyle{p}^{{m}}\) where p is prime number and m is a positive integer. Show that G contains an element of order p.
asked 2021-02-25
Let (Z,+) be a group of integers and (E,+) be a group of even integers. Find and prove if there exist an isomorphism between them.
asked 2021-02-08
Prove that if "a" is the only elemnt of order 2 in a group, then "a" lies in the center of the group.
asked 2021-02-21
How do you solve this problem? I don' t even know whereto begin.
A Ferrari with a mass of 1400 kg approaches a freeway underpassthat is 10 m across. At what speed must the car be moving, inorder for it to have a wavelength such that it might somehow"diffract" after passing through this "single slit"? How dothese conditions compare to normal freeway speeds of 30m/s?
asked 2020-11-09
Show that if \(\displaystyle{K}\subseteq{F}\) is an extension of degree k, every element \(\displaystyle\alpha\in{F}\) has a minimal polynomial over K of degree <=k. Find theminimal polymials of complex numers over \(\displaystyle\mathbb{R}\) explicitly to verify this fact
asked 2020-11-11
An nth root of unity epsilon is an element such that \(\displaystyle\epsilon^{{n}}={1}\). We say that epsilon is primitive if every nth root of unity is \(\displaystyle\epsilon^{{k}}\) for some k. Show that there are primitive nth roots of unity \(\displaystyle\epsilon_{{n}}\in\mathbb{C}\) for all n, and find the degree of \(\displaystyle\mathbb{Q}\rightarrow\mathbb{Q}{\left(\epsilon_{{n}}\right)}\) for \(\displaystyle{1}\le{n}\le{6}\)
asked 2021-02-27
Prove the following.
(1) Z ∗ 5 is a cyclic group.
(2) Z ∗ 8 is not a cyclic group.
asked 2021-01-25
Suppose G is a group and H is a normal subgroup of G. Prove or disprove ass appropirate. If G is cyclic, then \(\displaystyle\frac{{G}}{{H}}\) is cyclic.
Definition: A subgroup H of a group is said to be a normal subgroup of G it for all \(\displaystyle{a}\in{G}\), aH = Ha
Definition: Suppose G is group, and H a normal subgruop og G. THe froup consisting of the set \(\displaystyle\frac{{G}}{{H}}\) with operation defined by (aH)(bH)-(ab)H is called the quotient of G by H.
...