Cosets Definition and 66 Threads

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 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...