Is this what you meant to say? g to any integer power is an element of G because G is closed under multiplication.PsychonautQQ said:Homework Statement
if K is normal in G and has index m, show g^m is an element of K for all g in G
Work (I haven't done much with proofs so bear with me):
|G/K| = |G| / |K| = m
|G| = x
|K| = y
g^m must be an element of G since m|x
This is pretty much the definition of "normal subgroup". But you haven't even said, much less proved, that g is in K.if g^m is an element of G and K is normal to G then
(g^m)K = K(g^m) --> (g^m)K(g^m)^-1 = K for all g^m in G
is this work legit?
I don't see where this is taking you. Try using my hint to reword the problem in terms of the group ##G/K##, i.e. work with cosets of ##K## instead of individual elements of ##G##.PsychonautQQ said:I just saw your clue and will think about it and try to use it to help me figure the problem out, but meanwhile here is what I've done so far on my own.
|G| / |K| = m
|G| = x
|K| = y
x = my
G = {1,g,g^2...,g^(x-1)} are all elements of G so
{1,g,g^2...,g^(x-1)}^x are all elements of G so
{1,g,g^2...,g^(x-1)}^my are all elements of G
is this going in the right direction? I'll start working on this problem again now with the hint you've given
Yes, that's all there is to it.PsychonautQQ said:Solution: There are m cosets of K in G. This means that the group G/K has m elements in it, thus any element of that group raised to the m power would equal the identity element, K. (Kg)^m = Kg^m = K
Is this correct? If it is then wow,, I overthinking this
A factor group, also known as a quotient group, is a group that is formed by taking a subgroup of a larger group and "factoring out" the elements in that subgroup. This results in a new group with fewer elements that still retains some of the structure and properties of the original group.
The order of a factor group is determined by dividing the order of the original group by the order of the subgroup that is being factored out. This can be represented by the formula: Order of Factor Group = Order of Original Group / Order of Subgroup
Factor groups play an important role in group theory as they allow for a better understanding of the structure and properties of larger groups. They also help in simplifying complex groups and making them easier to study and classify.
Yes, factor groups can be isomorphic to each other. Isomorphic groups have the same structure and properties, even if their elements may be represented differently. This means that two factor groups formed by factoring out different subgroups from the same original group can still be isomorphic to each other.
Factor groups have applications in various fields such as cryptography, chemistry, and physics. In cryptography, factor groups are used in the creation of public-key encryption systems. In chemistry, factor groups are used to classify molecules based on their symmetry. In physics, factor groups are used in understanding the symmetries and conservation laws in physical systems.