Lemma of B. H. Neumann on coverings of a group by cosets. #
Let the group $G$ be the union of finitely many, let us say $n$, left cosets of subgroups $C₁$, $C₂$, ..., $Cₙ$: $$ G = ⋃_{i = 1}^n C_i g_i. $$
Subgroup.exists_finiteIndex_of_leftCoset_cover
at least one subgroup $C_i$ has finite index in $G$.Subgroup.leftCoset_cover_filter_FiniteIndex
the cosets of subgroups of infinite index may be omitted from the covering.Subgroup.exists_index_le_card_of_leftCoset_cover
: the index of (at least) one of these subgroups does not exceed $n$.Subgroup.one_le_sum_inv_index_of_leftCoset_cover
: the sum of the inverses of the indexes of the $C_i$ is greater than or equal to 1.Subgroup.pairwiseDisjoint_leftCoset_cover_of_sum_inv_index_eq_one
If the sum of the inverses of the indexes of the subgroups $C_i$ is equal to 1, then the cosets of the subgroups of finite index are pairwise disjoint.
A corollary of Subgroup.exists_finiteIndex_of_leftCoset_cover
is:
Subspace.union_ne_univ_of_lt_top
: a vector space over an infinite field cannot be a finite union of proper subspaces.
This can be used to show that an algebraic extension of fields is determined by the set of all minimal polynomials (not proved here).
[1] [Neu54], Groups Covered By Permutable Subsets, Lemma 4.1 [2] https://mathoverflow.net/a/17398/3332 [3] http://alpha.math.uga.edu/~pete/Neumann54.pdf
Let the group G
be the union of finitely many left cosets g i • H i
.
Then at least one subgroup H i
has finite index in G
.
Let the group G
be the union of finitely many left cosets g i • H i
.
Then the cosets of subgroups of infinite index may be omitted from the covering.
Let the group G
be the union of finitely many left cosets g i • H i
. Then the
sum of the inverses of the indexes of the subgroups H i
is greater than or equal to 1.
Let the group G
be the union of finitely many left cosets g i • H i
.
If the sum of the inverses of the indexes of the subgroups H i
is equal to 1,
then the cosets of the subgroups of finite index are pairwise disjoint.
B. H. Neumann Lemma : If a finite family of cosets of subgroups covers the group, then at least one of these subgroups has index not exceeding the number of cosets.