Let $G$ be a finite group or $|G|=n$ and let $(\phi(n) ,n)=1$ (where $\phi(n) $ is Euler's function). Now prove $G$ is abelian.
2025-01-13 02:20:32.1736734832
About Abelian Finite Groups and Euler's Function
228 Views Asked by amir bahadory https://math.techqa.club/user/amir-bahadory/detail At
1
There are 1 best solutions below
Related Questions in ABSTRACT-ALGEBRA
- Projective Indecomposable modules of quiver algebra
- Binary relations for Cobb-Douglas
- Relations among these polynomials
- Number of necklaces of 16 beads with 8 red beads, 4 green beads and 4 yellow beads
- Page 99 of Hindry's Arithmetics, follows from exact sequence that $\text{N}(IJ) = \text{N}(J)\text{card}(J/IJ)$?
- How to write the identity permutation as a product of transpositions
- Is $H$ a subgroup?
- $x=(0,\overline{1})$ and $y=(0,\overline{2})$ generate the same ideal in $R=\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$
- Having some problems with understanding conics and graphing (eccentricity)
- Is this Cayley Diagram contradictory?
Related Questions in FINITE-GROUPS
- Number of necklaces of 16 beads with 8 red beads, 4 green beads and 4 yellow beads
- Prove a subgroup is normal
- Is a finite group $G$ determined by the sequence $p(G,k)$ of probabilities that $G$ is generated by $k$ random elements?
- Normal subgroup of prime order in the center
- Order of subgroups formed by elements whose order divides a prime power
- Cardinality of a conjugacy class
- Order of elements in a cyclic group ($\mathbb Z_{26}$)
- commutator subgroup of upper triangular matrix
- In what sense are the linear characters among the irreducible characters
- Proof that the induced class function $\theta^G$ is a character if $\theta$ is a representation on subgroup
Related Questions in ABELIAN-GROUPS
- Obstruction to be conjugated by an automorphism for subgroups of an abelian group
- About Abelian Finite Groups and Euler's Function
- How can I show that the following pairs of permutations are in the same conjugacy class in S5:
- On the order of elements of the generating set?
- Showing an abelian group indecomposable
- Abelian group of order pq, two different prime numbers
- A group is a $\mathbb Z_n$-module if and only if the order of every element in the group divides n
- Does $A\oplus \mathbb{Z}\cong B\oplus \mathbb{Z}$ imply $A\cong B$?
- $H, N$ subgroups of $S_{5}$
- Proving a 1 Dimensional Lie Algebra is Abelian
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity
Every group $G$ of order $n$ such that $(n,\phi(n))=1$ is in fact cyclic, hence abelian. For an elementary proof see one of the following references (the first proof is simple enough to be suitable for an elemenatry class in group theory, the author says):
Jungnickel, Dieter. On the Uniqueness of the Cyclic Group of Order $n$. Amer. Math. Monthly, Vol. 99, No. 6 (1992)
Gallian, J. A. Moulton, David. When is $\mathbb{Z}_n$ the only group of order $n$?, Elemente der Mathematik, Vol. 48 (1993).