Testing whether a binary structure is a group

  • #1
1,462
44

Homework Statement


Consider the binary structure given by multiplication mod 20 on {4, 8, 12, 16}.
Is this a group? If not, why not?

Homework Equations




The Attempt at a Solution


I started by constructing a Cayley table, and working things out. It turns out that 16 acts as an identity element, 4 is the inverse of itself, 12 and 8 are mutual inverses, and 16 is the inverse of itself. One more things to check would be to see if the associative property is satisfied for all elements. However, this would seem to be a very tedious process.

On the other hand, I know that, up to isomorphism, there are only two types of groups of order 4, the cyclic group ##\mathbb{Z}_4## and the Klein four-group. Just by comparing tables, the Cayley table for this binary structure is equivalent to that of the Klein four-group. So is it valid to say, by isomorphism, that this binary structure is also a group, or do I have to explicitly show associativity?
 

Answers and Replies

  • #2
15,421
13,455
The isomorphism is sufficient, because you already know, that the other group is associative. Beside that, you only need the closure, i.e. that the multiplication stays inside the set. Associativity is then inherited by ##\mathbb{Z}_{20}## or even by ##\mathbb{Z}## itself.
 
  • Like
Likes Mr Davis 97

Related Threads on Testing whether a binary structure is a group

  • Last Post
Replies
3
Views
1K
Replies
14
Views
4K
Replies
3
Views
3K
  • Last Post
Replies
19
Views
3K
  • Last Post
Replies
8
Views
2K
Replies
7
Views
2K
  • Last Post
Replies
3
Views
2K
Replies
2
Views
1K
  • Last Post
Replies
1
Views
3K
Replies
2
Views
2K
Top