Basic Group Theory: Proof <A,B>n<C>=<AuBnC>

In summary, it is not always true that <A,B>n<C> = <AuB>n<C> in a group G, where A, B, and C are sets in G and <D> denotes the smallest subgroup in G containing the set D. This can be seen in the example of additive groups generated by {3} and {4}, where their intersection is empty. However, it is true that if A and B are subgroups of G, then <A,B> = AuB. This can be seen in the example given, where the left-hand side is a group while the right-hand side is not.
  • #1
tgt
522
2
In a group G, is it true that <A,B>n<C>=<AuBnC> where A,B and C are sets in G?

Where <D> denotes the smallest subgroup in G containing the set D.

Proof
If g is in <A,B> and g is in <C> then g is capable of being generated by elements in A or B and also elements in C. So g is generated by elements in (AuB)nC. So g is in <(AuB)nC>.

if g is in <(AuB)nC> then g is in <AuB> and g is in <C> so g is in <AuB>n<C>
 
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  • #2
tgt said:
If g is in <A,B> and g is in <C> then g is capable of being generated by elements in A or B and also elements in C. So g is generated by elements in (AuB)nC. So g is in <(AuB)nC>.
I don't think that is true. For example, consider the additive groups generated by {3} and by {4} (i.e. 3Z and 4Z).
They both contain 12, yet the intersection of the generating sets is empty.
 
  • #3
CompuChip said:
I don't think that is true. For example, consider the additive groups generated by {3} and by {4} (i.e. 3Z and 4Z).
They both contain 12, yet the intersection of the generating sets is empty.

nice one.
 
  • #4
We can say that if A and B are subgroups of G then <A,B>=AuB, right?
 
  • #5
Did you check with my example?

(in fact, the LHS is a group while the RHS is not).
 
  • #6
CompuChip said:
Did you check with my example?

(in fact, the LHS is a group while the RHS is not).

I see.
 

FAQ: Basic Group Theory: Proof <A,B>n<C>=<AuBnC>

What is basic group theory?

Basic group theory is a branch of mathematics that studies the properties and structure of groups, which are sets of elements with a binary operation that satisfies certain axioms. It is an important tool in abstract algebra and has applications in various fields such as physics, chemistry, and computer science.

What is a proof in basic group theory?

A proof in basic group theory is a logical argument that uses the axioms and definitions of groups to demonstrate the validity of a statement or theorem. It involves breaking down the statement into simpler components and using rules of inference to show that it is true.

What does the notation n mean?

The notation n represents the subgroup generated by the elements A and B in the group C. This subgroup consists of all possible combinations of A and B using the group operation, as well as their inverses. In other words, it is the smallest subgroup of C that contains both A and B.

How is the proof n= used in basic group theory?

The proof n= is used to show that the subgroup generated by the elements A and B is equal to the subgroup generated by their union in the group C. This result is important in understanding the structure of subgroups and their relationship to the original group.

Can you explain the concept of isomorphism in basic group theory?

In basic group theory, an isomorphism is a mapping between two groups that preserves the group structure. This means that if two groups are isomorphic, they have the same number of elements and the same group operation, and their elements can be paired with each other in a way that preserves the group axioms. Isomorphisms are important in understanding the similarities and differences between different groups.

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