Finite Index of Subgroup of Subgroup

Prove the following: If $H$ is a subgroup of limited index in a team $G$, and also $K$ is a subgroup of $G$ having $H$, after that $K$ is of limited index in $G$ and also $[G:H] = [G:K][K:H]$.

So this is primarily a bijective evidence? The variety of cosets of $H$ in $G$ amounts to the variety of cosets of $K$ in $G$ times the variety of cosets of $H$ in $K$ by the reproduction concept?

Reference: Fraleigh p. 103 Question 10.35 in A First Course in Abstract Algebra

2019-05-18 21:53:54
Source Share
Answers: 2

The approved map $G/H \to G/K$ is surjective. The fiber of $gK$ is $\{gkH : k \in K\}$, which can be understood $K/H$.

2019-05-21 07:26:55

In reality, the outcome holds also if you do not think that we are managing limited index, given you take the equal rights to be an equal rights of cardinalities. Simply adhere to the suggestion Pete Clark recommends, yet without thinking that the variety of cosets is limited in either instance. And also bear in mind that the variety of distinctive cosets of a subgroup $B$ in a team $A$ is equivalent, necessarily, to the index of $B$ in $A$, $[A:B]$.

Regarding whether the $x_i$ are "coset reps for $H$", bear in mind that for a collection of components to be "a set of coset reps" we call for that if $x_iH = x_jH$, after that $i=j$, which for every single $g\in G$ there exists $i$ such that $gH = x_iH$ (reasoning left cosets ; appropriate cosets function similarly). So $x_1,\ldots,x_n$ will certainly not usually be "a coset rep" neither a set of coset reps, yet that is immaterial. Merely show that if $\{x_i\}_{i\in I}$ is a set of coset reps for $K$ in $G$ (definition, (i) for every single $g\in G$ there exists $i\in I$ such that $gK = x_iK$ ; and also (ii) for all $i,i'\in I$, if $x_iK = x_{i'}K$ after that $i=i'$), and also $\{y_j\}_{j\in J}$ is a set of coset reps for $H$ in $K$ (definition, (1) for every single $k\in K$ there exists $j\in J$ such that $kH = y_jH$ ; and also (2) for all $j,j'\in J$, if $y_jH = y_{j'}H$, after that $j=j'$) ; after that it adheres to that $\{x_iy_j\}_{(i,j)\in I\times J}$ is a set of coset reps for $H$ in $G$ (definition, you require to confirm that (I) for all $g\in G$ there exists $(i,j)\in I\times J$ such that $gH = (x_iy_j)H$ ; and also (II) for all $(i,j),(i',j')\in I\times J$, if $(x_iy_j)H = (x_{i'}y_{j'})H$, after that $(i,j)=(i',j')$).

A reality that will certainly no question serve is to bear in mind that for any kind of team $A$ and also any kind of subgroup $B$ of $A$, $cB = dB$ if and also just if $cB\cap dB\neq\emptyset$.

2019-05-21 05:27:29