Intereting Posts

Arc length parameterization lying on a sphere
Why are so many of the oldest unsolved problems in mathematics about number theory?
Proving : $A \cap (B-C) = (A \cap B) – (A \cap C)$
Metric Space and Uniformly Continuous Functions
matrix derivative of gradients
How many cards do you need to win this Set variant
Classifying Types of Paradoxes: Liar's Paradox, Et Alia
What is the average rotation angle needed to change the color of a sphere?
A formal proof required using real analysis
Number of real roots of $2 \cos\left(\frac{x^2+x}{6}\right)=2^x+2^{-x}$
Coordinate-free notation for tensor contraction?
How to show that $2730\mid n^{13}-n\;\;\forall n\in\mathbb{N}$
Convergence of $\int f dP_n$ to $\int f dP$ for all Lipschitz functions $f$ implies uniform integrability
Can a nice enough ODE always be extended to the complex plane?
Numbers $n$ with $n,n+2$ coprime to $p_k\#$ on $$

Show that every group of prime order is cyclic.

I was given this problem for homework and I am not sure where to start. I know a solution using Lagrange’s theorem, but we have not proven Lagrange’s theorem yet, actually our teacher hasn’t even mentioned it, so I am guessing there must be another solution. The only thing I could think of was showing that a group of prime order $p$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. Would this work?

Any guidance would be appreciated.

- Two exercises on characters on Marcus (part 1)
- Does validity of Bezout identity in integral domain implies the domain is PID?
- Basic Subgroup Conditions
- Finding the ring of integers of $\mathbb Q$ with $\alpha^5=2\alpha+2$.
- Showing that the direct product does not satisfy the universal property of the direct sum
- If $M$ is an artinian module and $f : M\to M$ is an injective homomorphism, then $f$ is surjective.

- Show $R \setminus S$ is a union of prime ideals
- Is a faithful representation of the orthogonal group on a vector space equivalent to a choice of inner product?
- Applications of Abstract Algebra to elementary mathematics
- Show that there exists a $3 × 3$ invertible matrix $M$ with entries in $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = I_3$.
- How to prove that a complex number is not a root of unity?
- A question on coalgebras(1)
- Universal properties (again)
- Automorphism on integers
- If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
- Does $A^{-1}A=G$ imply that $AA^{-1}=G$?

As Cam McLeman comments, Lagranges theorem is considerably simpler for groups of prime order than for general groups: it states that the group (of prime order) has no non-trivial proper subgroups.

I’ll use the following

## Lemma

Let $G$ be a group, $x\in G$, $a,b\in \mathbb Z$ and $a\perp b$. If $x^a = x^b$, then $x=1$.

Proof: by Bezout’s lemma, some $k,\ell\in\mathbb Z$ exist, such that $ak+b\ell=1$. Then

$$ x = x^{ak+b\ell} = (x^a)^k \cdot (x^b)^\ell = 1^k \cdot 1^\ell = 1 $$(If you know a little ring theory, you might prefer to notice that the set $\{i | x^i=1\}\subseteq \mathbb Z$ forms an ideal which must contain $(a,b)=1$ if it contains $a$ and $b$.)

Now let $P$ be an arbitrary group of prime order $p$. Consider any $x\in P$ such that $x\neq 1$ and consider the set

$$ S = \{ 1, x, x^2 , \dots , x^{p-1} \}\subseteq P.$$

First assume two of these elements are equal, say $x^u=x^v$ and $u<v$ without loss of generality. Then $x^{v-u}=1$ and $1\leq v-u \leq p-1$. But then surely $v-u \perp p$. By the lemma, $x^{v-u} = x^p = 1$ now implies that $x=1$, a contradiction so every two members of $S$ must be different.

But then $|S|=p$. This implies $S=P$ and $P=\langle x\rangle$ is cyclic.

This is a proof of Cauchy’s theorem that does not use Lagrange, it is due to James McKay. It has an uncanny similarity to the proof of Fermat’s Little theorem using necklaces.

Let $G$ be a group of order $np$. Then there are $(np)^{p-1}$ solutions to $g_1g_2\dots g_p=1$ since for any values of $g_1,g_2,\dots ,g_{p-1}$ there is a unique inverse for $g_1g_2\dots g_{p-1}$. We call $S$ the set of solutions, we have asserted $|S|$ is a multiple of $p$.

Notice if $g_1,g_2\dots g_p=1$ then $g_ig_{i+1}\dots g_pg_1\dots g_{i-1}=1$ also.

Divide $S$ in rotation classes. Where $s$ is in the same class as $s’$ only if they are rotations of each other. Notice all classes have size $1$ or $p$ (this uses $p$ is prime).Therefore the number of classes of size $1$ is multiple of $p$. Since $\underbrace{1,1\dots ,1}_\text{p times}$ makes up a class of size $1$ there must be another, this provides the desired element of order $p$.

We may now use Cauchy Theorem to determine a group $G$ of order $p$ has an element $g$ of order $p$, this element generates a cyclic subgroup of order $p$. This subgroup must be $G$ so $G$ is cyclic.

The answer is fairly simple once Lagrange’s Theorem is quoted. We have no proper subgroups of smaller order. We only need to prove the uniqueness of the group of that size. For this note that given any element of such a group, continue to take powers of it … This series $x^r$ has to terminate because of closure. The series also has to exhaust all the elements of the group, otherwise we will have subgroups of a smaller order.

Thus we have proven that every group of prime order is necessarily cyclic. Now every cyclic group of finite order is isomorphic to $\mathbb{Z}_n$ under multiplication, equivalently, the group of partitions of unity of order $|G|$. Thus the uniqueness is proved.

- Ways to create a quadrilateral by joining vertices of regular polygon with no common side to polygon
- If $\mathop{\mathrm{Spec}}A$ is not connected then there is a nontrivial idempotent
- Initial Value Problem
- Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)
- Correlation between two linear sums of random variables
- test convergence of improper integrals 4
- compactness / sequentially compact
- Another integral for $\pi$
- The theory of discrete endless orders is complete
- An example of prime ideal $P$ such that $\bigcap_{n=1}^{\infty}P^n$ is not prime
- Criteria for swapping integration and summation order
- Show that $\prod (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$
- Learning mathematics as if an absolute beginner?
- What is $f(f^{-1}(A))$?
- An element is integral iff its minimal polynomial has integral coefficients.