Cosets and Lagrange’s Theorem (10/28)
• Definition. If H is a subgroup of G and if a
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G, then
the left coset of H containing a, denoted aH, is simply
{ah | h H}. Parallel definition for right coset.
If G is abelian, we can just say “coset”. Also, if G is an
additive group, we write a + H.
Note that H (= eH) itself is always one of its cosets.
Example. In Z, what are the cosets of H = 5Z?
Example. In Z12, what are the cosets of H = 4 ?
Example. In D4, what are the left cosets of V ?
What are the right cosets of V ?
Example. In D4, what are the left cosets of R180 ?
What are the right cosets of R180 ?
Key Role of Cosets
• Theorem. The cosets (left or right) of H in G always
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“partition” G, i.e., they are pairwise-disjoint, and their
union is all of G.
Theorem. If H is finite, then |aH| = |H| for all a G.
So the cosets of H :
• cover all of G,
• never overlap with each other,
• and all have the same order.
Check these on previous examples.
This leads us to one of the central results of finite group
theory:
Lagrange’s Theorem
• Theorem. If G is a finite group and if H is a subgroup of G,
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then |H| divides |G|.
For what class of groups did we already know this to be
true?
Note that if G and H are as in the theorem, then |G| / |H|
is just the number of cosets of H in G.
More generally (since the following definition can apply to
infinite groups also) the index of H in G, denoted [G:H], is
the number of cosets of H in G.
Example. What is the index of 5Z in Z?
Example. What is the index of V in D4?
Is the converse of Lagrange true?
• Theorem???? If the number m divides |G|, does G then
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have a subgroup H of order m?
Again, we know this (and more!) to be true about a
particular class of groups, right?
But, alas, it is not true in general. A4 provides a counterexample, and in fact is the smallest group to do so.
In general, it turns out that for n > 3, An does not contain
a subgroup of order n! / 4.
By a more advanced set of theorems,
the Sylow Theorems (Chapter 24), if m = pk,
i.e., if m is a power of a prime, then the Lagrange
converse does hold.
Nice corollaries of Lagrange
• Theorem. If a
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G, |a| divided |G|.
Theorem. Every group of prime order is cyclic.
Theorem. If a G, a|G| = e.
Fermat’s Little Theorem. For every a Z and
every prime p, a p mod p = a mod p.
Example. What is 540 mod 37?
We prove Fermat’s Little Theorem in MA 214
(Number Theory), but the proof is a little tricky and we
can’t use this proof since “we” don’t know any group
theory there.
Assignment for Wednesday
• Read pages 147-149.
• Finish up Exercises 1-9 on page 156 and also do
Exercises 14, 15, 16, 17, 18, 19, 22, and 23.