(4) Cosets to Index of a Subgroup Flashcards
What is the left of coset of H in G containing a?
- aH = {ah|h∈H}.
- equivalence class of a [a] with respect to ~L.
What is the right of coset of H in G containing a?
- Ha = {ha|h∈H}.
- equivalence class of a [a] with respect to ~R.
T/F. The sets of left together with right cosets of H in G forms a partition of G.
F. The sets of left or right
The left and right cosets of subgroup H are always subgroups of G.
T. e.g. 1+4Z
If G is abelian then
aH=Ha for every a in G
Let H be a subgroup of G of order n. What is the order of the cosets of H in G?
n
T/F. |aH|=|H|=|Ha|
T
Theorem of Lagrange
If H is a subgroup of a finite G. Then the order of H is a divisor of the order of G.
Let H be a subgroup of a finite group G. How many left cosets does H have in G?
|G|/|H|
The converse of the theorem of Lagrange is true iff
G is cyclic
Every group of prime order is cyclic. Why?
By theorem of Lagrange, the order of the non trivial subgroups of G will be equal to the order of G.
Let G be a group of order n where n is prime. How many generators does G have?
All non-identity elemets, n-1
Let G be a finite group of order n, a ∈ G. Then |a| divides n and ____
a^n=e
One group of prime order p up to isomorphism
Zsubp
T/F. The number of left cosets of H in G is always equal to the number of right cosets of H in G.
T
