Non-Abelian group of Order 12

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In summary: This will show that the group is not commutative, and therefore, not abelian. In summary, we showed that Z/3 X|\alpha Z/4 is a non-abelian group of order 12, and it is not isomorphic to A4 or D6 by examining its operation and order.
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Thank you for your time!

Homework Statement


Prove that Z/3 X|[tex]\alpha[/tex] Z/4 is a non-abelian group of order 12 which it is not isomorphic to A4 or to D6

X|][tex]\alpha[/tex] is supposed to be the sign for semi direct product.

Homework Equations



Z/3 X|[tex]\alpha[/tex] Z/4=<a,b / a4=b3=1, aba=a>
A4=<a,b / a3=b2=1, aba=ba-1b>
D6=<a,b / a2=b2=1, bab-1=a-1>


The Attempt at a Solution



I am just having issues wrapping my head around Z/3 X|[tex]\alpha[/tex] Z/4, I read somewhere that Z/3 X|[tex]\alpha[/tex] Z/4 = {x,y / x[tex]\in[/tex]Z/3 and y[tex]\in[/tex]Z/4} with the operation (x1,y1)(x2,y2)=...and that's where I am stuck.
 
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Can someone please explain this operation and how to prove Z/3 X|\alpha Z/4 is a non-abelian group of order 12 which it is not isomorphic to A4 or to D6. First, we need to show that Z/3 X|\alpha Z/4 is a group. We can do this by showing that it is closed under the operation of semi direct product, i.e. (x1,y1)(x2,y2)=(x1x2,\alpha(x1)(y2)) where x1,x2\inZ/3 and y1,y2\inZ/4. We can also show that the operation is associative and that it has an identity element (e,e) where e is the identity element in Z/3 and Z/4. Finally, we need to show that every element has an inverse. To do this, we need to find an inverse for each element (x,y) in the group. The inverse of (x,y) is (x-1,\alpha(x-1)(y-1)) where x-1 and y-1 are the inverse elements in Z/3 and Z/4, respectively.Now, we need to show that Z/3 X|\alpha Z/4 is not isomorphic to A4 or D6. This can be done by examining the operations in each group and noting the differences. For example, A4 has the operation aba=ba-1b while Z/3 X|\alpha Z/4 has the operation aba=a. Similarly, D6 has the operation bab-1=a-1, while Z/3 X|\alpha Z/4 does not have such an operation. We can also show that Z/3 X|\alpha Z/4 is not isomorphic to A4 or D6 by examining the orders of each group. A4 has order 12, while D6 has order 12. Since Z/3 X|\alpha Z/4 has order 12 as well, it cannot be isomorphic to either of these groups. Finally, we need to show that Z/3 X|\alpha Z/4 is a non-abelian group. We can do this by finding two elements in the group such that the
 

1. What is a non-Abelian group of order 12?

A non-Abelian group of order 12 is a mathematical structure consisting of 12 elements and a binary operation that is not commutative, meaning that the order in which elements are combined matters. In other words, the elements do not necessarily follow the commutative property of multiplication, where a x b = b x a.

2. How is a non-Abelian group of order 12 different from an Abelian group of the same order?

A non-Abelian group of order 12 differs from an Abelian group of the same order in that the elements in a non-Abelian group do not necessarily commute, while the elements in an Abelian group always commute. This means that the binary operation in a non-Abelian group is not commutative, while it is in an Abelian group.

3. What are some examples of non-Abelian groups of order 12?

Some examples of non-Abelian groups of order 12 include the dihedral group of order 12, the symmetric group of degree 4, and the quaternion group of order 8. These groups have different structures and properties, but all have 12 elements and a non-commutative binary operation.

4. What are the applications of non-Abelian groups of order 12?

Non-Abelian groups of order 12 have many applications in mathematics, physics, and computer science. In mathematics, they are used to study symmetry and group theory. In physics, they are used to describe the symmetries of physical systems. In computer science, they are used in cryptography and coding theory.

5. What are the properties of a non-Abelian group of order 12?

A non-Abelian group of order 12 has several important properties, including closure, associativity, identity element, inverses, and non-commutativity. It also has a group structure that can be represented by a Cayley table, which shows the result of combining any two elements in the group. Additionally, it has subgroups and cosets, which allow for the analysis of smaller groups within the larger group.

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