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math8
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If A and B are subgroups of G and AB=G does it follow that AB=BA? If yes, why?
math8 said:If A and B are subgroups of G and AB=G does it follow that AB=BA? If yes, why?
AB is the set of elements of the form ab (with a in A & b in B).sutupidmath said:What does AB mean, i mean what operation are you performing between A and B here? is this supposed to be the same operation of the group G, or?
HallsofIvy said:tgt, can you give an example in which AB= G and BA= {1}?
Your proof should have stopped here - this is all you need to show. AB=BA is an equality of sets.sutupidmath said:ok, so what about this.
Like said AB is the set of all elements such a is in A and b is in B. Then from this follows that also [tex] a^{-1}b^{-1}\in AB, since, a^{-1}\in A, b^{-1}\in B[/tex] this comes from the fact that both A,B are subgroups , so they do have inverses.
NOw since AB=G, it means also that the inverse of [tex] a^{-1}b^{-1}[/tex] is in AB. that is :
[tex] (a^{-1}b^{-1})^{-1}=ba \in AB[/tex]
This step is wrong. Can you see why?[tex]b^{-1}(ba)=a[/tex] now multiplying with b we get
[tex] ba=ab[/tex]
Hell YES! This happens when you assume that the same thing is going to happen in next step as well. We cannot do this, because we don't know whether A, B are commutative. so by doing what i did in the last step, i should have multiplied by b from the left side, which would get us nowhere, since we would end up with ba=ba... and this is not what we wanted.morphism said:This step is wrong. Can you see why?
Group theory is a branch of mathematics that studies the properties of groups, which are sets of elements that follow certain rules of composition. These rules define how the elements of a group can be combined and manipulated, and the study of group theory helps us understand the structure and behavior of these groups.
Group theory has many applications in various fields, including physics, chemistry, computer science, and cryptography. It is used to study symmetries in physical systems, analyze molecular structures, and design secure communication protocols, among other things.
The main concepts in group theory include group operations, subgroups, cosets, and group homomorphisms. Group operations define how elements in a group can be combined, while subgroups are smaller groups that are contained within a larger group. Cosets are sets of elements that are related to a subgroup, and group homomorphisms are functions that preserve the structure of a group.
Group theory is a fundamental part of modern mathematics, as it provides a powerful way to analyze symmetry and structure in various mathematical objects. It also has connections to other branches of mathematics, such as algebra, topology, and number theory, making it a crucial tool for understanding a wide range of mathematical concepts.
Yes, there are many different types of groups, each with its own properties and applications. Some common types of groups include cyclic groups, permutation groups, and matrix groups. There are also more specialized groups, such as Lie groups, which are used in physics and differential geometry.