Show that every proper subgroup of $S_3$ is cyclic. So I approached it like this, $$|S_3|=6$$ So divisors of 6 are 2 and 3 (excluding 1 and 6, because improper subgroups). Both 2 and 3 are prime and any group of prime order is cyclic. Is that a correct approach?
2026-04-05 13:47:40.1775396860
Show that every proper subgroup of $S_3$ is cyclic.
1.2k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ABSTRACT-ALGEBRA
- Feel lost in the scheme of the reducibility of polynomials over $\Bbb Z$ or $\Bbb Q$
- Integral Domain and Degree of Polynomials in $R[X]$
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- Group with order $pq$ has subgroups of order $p$ and $q$
- A commutative ring is prime if and only if it is a domain.
- Conjugacy class formula
- Find gcd and invertible elements of a ring.
- Extending a linear action to monomials of higher degree
- polynomial remainder theorem proof, is it legit?
- $(2,1+\sqrt{-5}) \not \cong \mathbb{Z}[\sqrt{-5}]$ as $\mathbb{Z}[\sqrt{-5}]$-module
Related Questions in GROUP-THEORY
- What is the intersection of the vertices of a face of a simplicial complex?
- Group with order $pq$ has subgroups of order $p$ and $q$
- How to construct a group whose "size" grows between polynomially and exponentially.
- Conjugacy class formula
- $G$ abelian when $Z(G)$ is a proper subset of $G$?
- A group of order 189 is not simple
- Minimal dimension needed for linearization of group action
- For a $G$ a finite subgroup of $\mathbb{GL}_2(\mathbb{R})$ of rank $3$, show that $f^2 = \textrm{Id}$ for all $f \in G$
- subgroups that contain a normal subgroup is also normal
- Could anyone give an **example** that a problem that can be solved by creating a new group?
Related Questions in CYCLIC-GROUPS
- Confusing step in proof of property of cyclic group automorphisms
- If $G=\langle x\rangle$ is cyclic group and order of $G$ is $40$ then how many order of $x^3$
- How to arrange $p-1$ non-zero elements into $A$ groups of $B$ where $p$ is a prime number
- $e^{n/e}$ estimate of the maximum order of permutation group element: proof
- Intuitive understanding of $g^iH=(gH)^i$ factor groups
- Exams exercise involving the permutation group $S_5$
- Find the order of 5 in $\mathbb Z_{12}$
- The commutator of two subgroup in a finite group
- Show that, for every $x\ \epsilon \ C_{m}$, we have that $ord(f(x))$ is a divisor of d.
- Why are $-1$ and $1$ generators for the Set of integers under addition?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- 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?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- 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
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
That's a correct approach, yes. I think that a more natural approach (but that's a matter of taste) would be to make the list of all proper subgroups of $S_3$ (there are only four such subgroups) and to check that each of them if cyclic.