I have a group $G$ and a random subgroup $H\leq G$. Also I have a homomorphism $\varphi\colon G\rightarrow H$. If $\lvert\ker(\varphi)\rvert=p$ and $p$ is prime can you help me how I can prove $G=\ker{(\varphi)}$? I try with the First Group Isomorphism Theorem but I could not do anything.
2026-03-25 14:33:15.1774449195
If $\lvert\ker(\varphi)\rvert=p$ and $p$ is prime, then $G=\ker(\varphi)$
50 Views Asked by user539638 https://math.techqa.club/user/user539638/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 PRIME-NUMBERS
- New prime number
- Confirmation of Proof: $\forall n \in \mathbb{N}, \ \pi (n) \geqslant \frac{\log n}{2\log 2}$
- How do I prove this question involving primes?
- What exactly is the definition of Carmichael numbers?
- I'm having a problem interpreting and starting this problem with primes.
- Decimal expansion of $\frac{1}{p}$: what is its period?
- Multiplying prime numbers
- Find the number of relatively prime numbers from $10$ to $100$
- A congruence with the Euler's totient function and sum of divisors function
- Squares of two coprime numbers
Related Questions in GROUP-ISOMORPHISM
- Symmetries of the Tetrahedron - Geometric description and isomorphic correlations
- Showing that $2$ of the following groups are not isomorphic
- When can the isomorphism theorem for Groups be rewritten as a direct product?
- Smallest $n\in \mathbb{Z}_{>0}$ for existence of a monomorphism $G \rightarrow S_n$
- $\mathrm{Hom}(\mathrm{Hom}(G,H),H) \simeq G$?
- Do the results hold for isomorphisms of groups?
- Isomorphism about direct product of multiplicative group and direct product of additive group
- Direct Sums of Abelian Groups/$R$-Modules
- Injective Morphisms of Modules and Bases
- Suppose$f:S_{3}\longrightarrow R^{\ast}$is Homomorphism.Then Kernal of h has
Related Questions in GROUP-HOMOMORPHISM
- homomorphism between unitary groups
- Order of a group = Order of kernel × Order of homomorphic image?
- Construct a non trivial homomorphism $\mathbb Z_{14} \to\mathbb Z_{21}$
- Continuous group homomorphism between normed vector spaces are linear?
- Show $\widehat{\mathbb{Z}}$ is isomorphic to $S^1$
- Coset and Fiber
- Finding a homormorphism form $\mathbb{Z}/4\mathbb{Z}$ to $\mathbb{Z}/6\mathbb{Z}$
- Show that the element $φ(a)\in G'$ has also order d!
- Explicit description of the group of homomorphisms from $\mathbb{Z}_p^{\times}$ to $\mathbb{Z}/n$
- Smallest $n\in \mathbb{Z}_{>0}$ for existence of a monomorphism $G \rightarrow S_n$
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?
This is not true, take $p:G\times G\rightarrow G$ the projection where the cardinal of $G$ is prime.