Understanding Isomorphic Groups in Group Theory

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In summary, group theory involves understanding different types of groups such as dihedral, cyclic, abelian, symmetric, and orthogonal groups. In order to determine if two groups are isomorphic, we look at their order and structure. In the specific example given, Z100 and Z2 x Z50 are isomorphic because they have the same order and are both cyclic, as well as D50 and Z4 x Z25 because they have the same order and structure.
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Sinico1234
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Hey trying to understand group theory a bit more. Need some help understanding how to spot which groups are isomorphic to what.

First of all, but silly, but confirmation on what the groups D is dihedral right? What sort of other groups with i have to deal with. Like Z for integers? and U for non-zero integers?

I know it has something to do with cyclic and abelian but i am still unsure.

One of the questions i have is given 4 grroups.

Z100, Z2 x Z50, D50, Z4 x Z25

which of the groups are isomorphic.

Any help is much appreciated.

Thanks
 
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!Yes, D is the dihedral group. Other groups you could encounter include C (cyclic group), A (abelian group), S (symmetric group), and O (orthogonal group). For the given question, Z100 and Z2 x Z50 are isomorphic since they are both cyclic groups of order 100. D50 and Z4 x Z25 are also isomorphic since they both have order 50.
 

1. What is an isomorphic group?

An isomorphic group is a mathematical concept in group theory where two groups have the same structure, meaning they have the same number of elements and the same way of combining those elements.

2. How do you prove that two groups are isomorphic?

To prove that two groups are isomorphic, you must show that there exists a one-to-one correspondence between the elements of the two groups and that this correspondence preserves the group operation. This can be done by constructing a bijective function between the two groups.

3. What is the significance of isomorphic groups in group theory?

Isomorphic groups play an important role in group theory because they allow us to study a particular group by understanding a more familiar group that is isomorphic to it. This can make complex groups easier to understand and classify.

4. Can two groups be isomorphic even if their elements are different?

Yes, two groups can be isomorphic even if their elements are different. Isomorphism is based on the structure of the group, not the specific elements. As long as the two groups have the same number of elements and the same way of combining them, they can be considered isomorphic.

5. Are all groups isomorphic to themselves?

Yes, all groups are isomorphic to themselves. This is because a group is isomorphic to itself if there exists a function that maps each element to itself, preserving the group operation. In other words, the identity function is always an isomorphism from a group to itself.

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