Order of Group Elements: Z3 x Z3 & Z2 x Z4

In summary, the question is about finding the order of elements in the groups (Z3 x Z3, +) and (Z2 x Z4, *) and (Z3 x Z5, *). In the first group, the maximum order is 3, and in the second group, the order of each element can be found by multiplying it by itself until it equals (1,1). However, the operation of multiplication does not make sense for these groups and it is likely a typo. In the third group, the order of elements (0,b) will be equal to the order of the "right" element, but multiplication is not a valid operation for this group.
  • #1
kljoki
5
0
Hi
i need a little help
i was given group (Z3 x Z3,+) and i should find order of every elements
so the elements are {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),( 2,2)} and the order of every element is
(0,0) has order 1
(0,1)*3=(0(mod 3),3(mod 3)) = (0,0) order 3
(0,2)*3=(0(mod 3),6(mod 3)) = (0,0) order 3
(1,0)*3=(3(mod 3),0(mod 3)) = (0,0) order 3
(1,1)*3=(3(mod 3),3(mod 3)) = (0,0) order 3
(1,2)*3=(3(mod 3),6(mod 3)) = (0,0) order 3
(2,0)*3=(6(mod 3),0(mod 3)) = (0,0) order 3
(2,1)*3=(6(mod 3),3(mod 3)) = (0,0) order 3
(2,2)*3=(6(mod 3),6(mod 3)) = (0,0) order 3

(a,b) + (a,b) + (a,b) = (3a(mod3), 3b(mod3))=(0,0) so max order is 3

next is group (Z2 x Z4, *)
the elements are {(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3)}
so here i should multiply every element n times till i get (an(mod 2),bn(mod 4)) = (1,1) so the order is n (i'm not sure about this correct me if I'm wrong)
the element (0,0) always have order one
and what about the other elements?

example the element (0,2)
there isn't ANY n with (0,2)n (mod 2, mod 4) = (1,1)
please help :)
thanks for your time
 
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  • #2
But [itex]\mathbb{Z}_2\times\mathbb{Z}_4[/itex] isn't even a group under multiplication. Are you sure they didn't say that the operation is addition?
 
  • #3
micromass said:
But [itex]\mathbb{Z}_2\times\mathbb{Z}_4[/itex] isn't even a group under multiplication. Are you sure they didn't say that the operation is addition?

On top of that, [itex]\mathbb{Z}_4[/itex] itself isn't a group under multiplication either. At best, it could be a field (if 4 were prime), so talking about a product like [itex](\mathbb{Z}_2\times\mathbb{Z}_4,*)[/itex] doesn't even make sense. This has to be a typo.
 
Last edited:
  • #4
micromass said:
But [itex]\mathbb{Z}_2\times\mathbb{Z}_4[/itex] isn't even a group under multiplication. Are you sure they didn't say that the operation is addition?
they say to try with multiplication to see what is going to happen
 
  • #5
thanks for the answer and explanation :)
 
  • #6
ok and what about (Z3xZ5,*)
(0,0),(0,1),(0,2),(0,3),(0,4) will have order 1 or what??
how can (an(mod 3),bn(mod 5)) = (1,1) when the element have (0,b)?? :D
what should i do here??
 
Last edited:
  • #7
These will have order equal to the order of the "right" element. This is not a group under multiplication (see my earlier edit - I was mistaken. Zero never has a multiplicative inverse.); only addition.
 

1. What is the significance of the order of group elements Z3 x Z3 & Z2 x Z4?

The order of group elements Z3 x Z3 & Z2 x Z4 is important because it indicates the number of elements in the group and the structure of the group. It also determines the properties and behaviors of the group elements.

2. How do you calculate the order of group elements Z3 x Z3 & Z2 x Z4?

The order of a group is equal to the number of elements in the group. For Z3 x Z3, the order would be 3 x 3 = 9. For Z2 x Z4, the order would be 2 x 4 = 8. The total order of the group would be the product of these two orders, which is 9 x 8 = 72.

3. What is the difference between Z3 x Z3 and Z2 x Z4?

Z3 x Z3 and Z2 x Z4 are both groups, but they have different structures. Z3 x Z3 is a cyclic group, meaning it has one generator that can create all other elements in the group. Z2 x Z4 is a direct product of two groups, meaning it is composed of two separate groups that do not interact with each other.

4. How are the elements in Z3 x Z3 & Z2 x Z4 represented?

The elements in Z3 x Z3 & Z2 x Z4 are typically represented using a binary operation, such as addition or multiplication. For example, in Z3 x Z3, the elements would be represented as (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2). In Z2 x Z4, the elements would be represented as (0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (1,3).

5. What are the applications of Z3 x Z3 & Z2 x Z4 in science?

Z3 x Z3 & Z2 x Z4 have various applications in science, particularly in the fields of mathematics and physics. They are used to study and understand group theory, which has applications in abstract algebra, cryptography, and coding theory. In physics, these groups are used to study symmetry and conservation laws in systems such as crystals and molecules.

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