Intereting Posts

Solving complex numbers equation $z^3 = \overline{z} $
Zeros of a Tropical polynomial
Binary operation (english) terminology
mathematical books
What is the quotient of a cyclic group of order $n$ by a cyclic subgroup of order $m$?
Is there a vector space that cannot be an inner product space?
Show that if $x,x^{'}\in C$ then $C_x=C_{x^{'}}$.
Birational and faithfully flat $\implies$ isomorphism
What makes Probability so difficult to get it right in the first place?
Evaluating the integral $\int_0^\infty \frac{\sin x} x \ dx = \frac \pi 2$?
A first order sentence such that the finite Spectrum of that sentence is the prime numbers
Confusion about Homotopy Type Theory terminology
An application of J.-L. Lion's Lemma
On integral of a function over a simplex
Understanding Strategy:Minimum number of weighing to find the faulty bag

Suppose $G$ is finite and $G=H\cup K\cup L$ for proper subgroups $H,K,L$. Show that $|G:H|=|G:K|=|G:L|=2$.

What I did: so if some of $H,K,L$ is contained in another, then we have $G$ being a union of two proper subgroups, which is impossible due to another result. So none of $H,K,L$ is contained is each other. If some element $h$ belongs to only $H$ and $k$ belongs to only $K$, then its product $hk$ cannot be in $H$ or $K$, so must belong to only $L$.

- Enumerating Bianchi circles
- Minimum number of out-shuffles required to get back to the start in a pack of $2n$ cards?
- Showing the countable direct product of $\mathbb{Z}$ is not projective
- Group homomorphisms and which of the following statements are true (NBHM-$2014$)
- Reconciling Different Definitions of Solvable Group
- Torsion Subgroup (Just a set) for an abelian (non abelian) group.

- If $f(x,y)\in R$ is an irreducible polynomial, is $R/f(x,y)$ a field?
- A subgroup such that every left coset is contained in a right coset.
- Prove that $\Bbb{Z}/I$ is finite where I is an ideal of $\Bbb{Z}$
- Prove that any finite group $G$ of even order contains an element of order $2.$
- Number of permutations for a cycle-type
- Is Pythagoras the only relation to hold between $\cos$ and $\sin$?
- Is it coherent to extend $\mathbb{R}$ with a reciprocal of $0$?
- Finiteness of the Algebraic Closure
- Characterizing units in polynomial rings
- Is the torsion-free subset not always a subgroup?

At least one of $H, K$ and $L$ must have index $2$ in $G$ by counting. Let’s say $|G:H| = 2$. Then, $H$ is normal in $G$. Then, $|K:K\cap H| = 2 = |L:L\cap H|$.

Now, for $G = H \cup (K – K\cap H) \cup (L – L\cap H)$, this union must be disjoint by counting and we must have $|G:K| = 2 = |G:L|$.

- Let $\{a_n\}$ be defined as follows: $a_1 = 2$, $a_{n+1}=\frac{1}{3-a_n}$, if $n \geq 1$. Does $\{a_n\}$ converge?
- Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
- Question on the symmetric product $\mathrm{Sym}^g\Sigma$
- Hahn-Banach theorem (second geometric form) exercise
- Why Are the Reals Uncountable?
- Why $E$ is the algebraic closure of $K$?
- Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$
- Solving 5th degree or higher equations
- Dual space of Bochner space
- Explicit formula of Mobius Transformation that maps non intersecting circles to concentric circles
- Characterising reals with terminating decimal expansions
- Is there a clever solution to this elementary probability/combinatorics problem?
- Combinatorics/variation dinner problem
- Solutions of $\prod_{i=1}^n x_i = c$ mod p
- Example of a closed subspace of a Banach space which is not complemented?