Alphabeta Math

Cosets and Lagrange's Theorem: Examples and Counterexamples

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✓ 8 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

For n≥1, the cosets of nℤ are the n congruence classes modulo n

Example

[G:G]=1 and, for finite G, [G:{e}]=|G|

Example

The subgroup orders in Sym({1,2,3}) are 1,2,3 and 6

Counterexample

A left coset that is not the corresponding right coset in Sym({1,2,3})

Example

Every positive divisor of the order of a finite cyclic group occurs as the order of a subgroup

Counterexample

2ℤ has index 2 in ℤ and is nevertheless equinumerous with ℤ

Counterexample

The product set HK of two subgroups need not be a subgroup

False statement

Every left coset of a subgroup is itself a subgroup

Sources