If we disregard direct products, we find that quaternion group is the only non-Abelian group which has every subgroup as normal. Is there some deep property responsible for this 'outstanding' property of the quaternion group? (ref)
2026-04-06 04:59:15.1775451555
What is so special about the quaternion group that it is the only non abelian group for which every subgroup of it is normal?
257 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated 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 NORMAL-SUBGROUPS
- subgroups that contain a normal subgroup is also normal
- Prime Ideals in Subrings
- Comparing centers of group and a subgroup
- Example for subgroups $H$ and $K$ where $HK = K H$ and neither $H$ nor $K$ is normal?
- How to show that every group in the normal series of $G$ is normal in $G$
- Lie groups with SU(12) as a subgroup
- determine if a subgroup of a free group is normal
- Is being a contranormally closed subgroup a transitive property?
- $G_{1}, G_{2} \triangleleft G$, $G_{1}G_{2} = G$, $G_{1} \cap G_{2} = \{ e \}$ implies $G_{1} \times G_{2} \cong G$?
- For prime $p$, normal subgroups of $SL(2, \mathbb Z/p\mathbb Z)$ remains normal in $GL(2, \mathbb Z/p\mathbb Z)$?
Related Questions in QUATERNIONS
- Intuition behind quaternion multiplication with zero scalar
- Universal cover $\mathbb{S}^3 \rightarrow SO(3)$ through Quaternions.
- Variance of a set of quaternions?
- Finding the Euler angle/axis from a 2 axes rotation but that lies on the original 2 axes' plane
- How many different quaternions $q$ are in a satisfying equation $q^2 = 1$?
- Dual quaternions displacement
- Why quaternions is a group?
- Why does the real part of quaternion conjugation with a pure quaternion stay 0?
- Why does the multiplication in a division algebra depends on every component?
- derive quaternion from rotation matrix, via eigenvector
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?