In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does H. Furthermore, H itself is both a left coset and a right coset. The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by [G : H].
Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group G, the number of elements of every subgroup H of G divides the number of elements of G. Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes.
I am having trouble understanding this example:
Let G=S_3 and H={(1),(13)}. Then the left cosets of H in G are
(1)H=H
(12)H={(12), (12)(13)}={(12),(132)}=(132)HI cannot figure out how to produce this relation:
(12)H={(12), (12)(13)}={(12),(132)}=(132)H
I understand (12)H={(12), (12)(13)} but...
This is a proof I am struggling on ...
Let H be a subgroup of the permutation of n and let A equal the intersection of H and the alternating group of permutation n. Prove that if A is not equal to H, than A is a normal subgroup of H having index two in H.
My professor gave me the hint to...
G group, H subgroup of G.
Suppose aH and bH are distinct leftcosets then Ha and Hb must be distinct right cosets?
My humble thoughts:
the left coset aH consists of a times everything in H;
Ha consists of everything in H times a.
Then this argument above is true?
Question:
Prove the following properties of cosets.
Given:
Let H be a subgroup and let a and b be elements of G.
H\leq\ G
Statement:
aH=bH \ if\ and\ only\ if\ a^{-1}b\ \epsilon\ H
The statement is what I have to prove.
My issue is I don't know how to start off the problem. When I...
Homework Statement
Let H be a subgroup of G such that g^-1hg is an element of H for all g in G and all h in H. Show that every left coset gH is the same as the right coset Hg.
Homework Equations
The Attempt at a Solution
need to show gh1=h2g
I know I need to show this, but am...
Homework Statement
In each case find the right and left cosets in G of the subgroups H and K of G.
a) G = A4; H = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, K = <(1 2 3)>
b) G= Z12; H = 3Z12, K = 2Z12
c) G = D4 = D3 = {1,a,a2
, a3, b, ba, ba2, ba3}, |a|=4,|b|=2, and aba = b; H =<a2>, K=<b>...
Homework Statement
suppose that H and K are subgroups of a group G such that K is a proper subgroup of H which is a proper subgroup of G and suppose (H : K) and (G : H) are both finite. Then (G : K) is finite, and (G : K) = (G : H)(H : K).
**that is to say that the proof must hold for...
Homework Statement
What is the stabilizer of the coset aH for the operation of G on G/H
The Attempt at a Solution
Its hard for me to do this because i don't really understand the problem. i know that the stabalizer of an ELEMENT s in some group is the subgroup Gs = {g element of...
[SOLVED] General question on cosets
Assume a group G is finite, abelian. Let p^n be a divisor of |G|, p prime, n >1. Let x in G have ord(x) = p (by Cauchy's theorem).
Here's my question. If you now take the factor group of G/<x> you have an element of that factor group that is also of...
Homework Statement
Let H be a subgroup of a finite group G. I understand that the cosets of H partition G into equivalence classes. Is it always true that each of these equivalence classes is a group?
EDIT: clearly is it not always true; let H ={0,4,8,12} in Z_16 and take the right coset with...
Homework Statement
Prove that if H is a subgroup of a finite group G, then the number of right cosets of H in G equals the number of left cosets of H in G
Homework Equations
Lagrange's theorem: for any finite group G, the order (number of elements) of every subgroup H of G divides...
Does cosets exist in rings?
i.e R = Ring, a in R
set {a*R}
or
set {a+R}
The above two sets looks very similar to cosets in groups but there are two operations in rings so potentially two different cosets both involving the same ring R and element a. If the above two sets are not...
"Let H and K be subgroups of a group G. Prove that the intersection xH\cap yK of two cosets of H and K is either empty or is a coset of the subgroup H\cap K."
I'm stuck here.
so let's say |a| = 30. How many left cosets of <a^4> in <a> are there?
ok, so |a| = 30. and think I need to find the order of <a^4> also. I thought the order of it is
<a^4> = e, a^4, a^8, a^12, a^16, a^20, a^24, a^28. so it has order of 8. but my book said the order of it is 15...
so I'm solving problems that tell me to find the left cosets, but I don't really know what they are.
by defn, let G be a group and H a subgp of G.. and let a be an element of G. the set ah for any h in H, denoted by aH is the left coset.
I mean, what does that mean. so for an example...
so let H = { 0, 3, 6} under Z(9), and I need to find a + H.
the book shows
0 + H = 3 + H = 6 + H
1 + H = 4 + H = 7 + H
2 + H = 5 + H = 8 + H.
I"m not understanding why they start with 0, 1, 2. what gives that away?
(G is a group, and H is a subgroup of G). I've just read in a book, that all distinct (left or right) cosets of H in G form a partition of G, i.e. that G is equal to the union of all those cosets. Apparently, this follows from the fact that two cosets are either equal or disjoint (I've proved...
Hello,
It should be common knowledge now that I have trouble with Group Theory. I would like to go back and start from the beginning but I haven't the luxury of time at this point. So for the present time I am resigned to just keeping up with the class the best I can. For anyone has the time...