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In a finite group, the subgroup, every coset and the set of cosets are finite

Statement

Let $G$ be a finite group and $H\le G$ . Then $H$ , every left and right coset of $H$ , and the coset set $G/H$ are finite. Moreover every coset has cardinality $|H|$ , and $[G:H]=|G/H|$ is a natural number.

Facts & Assumptions

Given: A finite group $G$ and a subgroup $H\le G$ .

[L1]

Every subset of a finite set is finite (A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$ , and equality holds if and only if $B = A$ ).

[L2]

The power set of a finite set is finite ( $\lvert\mathcal{P}(A)\rvert = 2^{\lvert A\rvert}$ for finite $A$ ).

[L3]

Every left or right coset of $H$ is equinumerous with $H$ (Every left or right coset of $H$ is equinumerous with $H$ ).

[F1]

A bijection transports finiteness and finite cardinality (The cardinality $\lvert A\rvert$ of a finite set).

[F2]

The coset set is $G/H=\{gH:g\in G\}$ , and its finite cardinality is the index (The coset set $G/H$ and the index $[G:H]$ of a subgroup, The left cosets of a subgroup partition the group).

Proof

technique · direct

1.1

Since $H\subseteq G$ and $G$ is finite, $H$ is finite by [L1].

givenL1

1.2

Every coset is a subset of $G$ , so $G/H\subseteq\mathcal P(G)$ . The power set is finite by [L2], hence $G/H$ is finite by [L1].

F2L1L2

2.1

Every coset is equinumerous with $H$ , so every coset is finite and has cardinality $|H|$ .

step 1.1L3F1

3.1

Therefore $[G:H]=|G/H|\in\mathbb N$ , and the finiteness and cardinality assertions are steps 1.1, 1.2 and 2.1.

step 1.1step 2.1step 1.2F2∎

Depends on

Used by

Lagrange's theorem: |G|=[G:H]|H| for every subgroup H of a finite group G Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 71 results over 26 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Thomas W. Judson, Abstract Algebra: Theory and Applications, Cosets and Lagrange's Theorem (standard reference, not scraped)
Thomas W. Judson, Abstract Algebra: Theory and Applications, §6.1: Cosets (standard reference, not scraped)
Thomas W. Judson, Abstract Algebra: Theory and Applications, §6.2: Lagrange's Theorem (standard reference, not scraped)