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rss1
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hi
Show that Z2 X Z3 IS ISOMORPHIC TO THE GROUP G = (1,2,4,8,10)
Show that Z2 X Z3 IS ISOMORPHIC TO THE GROUP G = (1,2,4,8,10)
rss said:hi
Show that Z2 X Z3 IS ISOMORPHIC TO THE GROUP G = (1,2,4,8,10)
I like Serena said:Hi rss! Welcome to MHB! (Smile)
Which group does G represent?
Either way, since Z2 x Z3 contains 6 elements, and G contains 5 elements, they cannot be isomorphic.
rss said:hi
this question is incorrect.
the correct question is
show that the group Z2xZ3 is isomorphic to the group G=(1,2,4,5,7,8) with respect to multiplication modulo 9
Euge said:Hi rss,
Consider the elements of order two in $\Bbb Z_2 \times \Bbb Z_3$ and $G$. If the groups don't have the same number of elements of order two, then they are not isomorphic.
rss said:there are a total of 6 elements in each group so they must be isomorphic.
rss said:there are a total of 6 elements in each group so they must be isomorphic. i just have to prove them so
Z2xZ3= (0,0),(0,1),(0,2),(1,0),(1,1),(1,2)
and G=(1,2,4,5,7,8)...0N MULTIPLICATION with respect to modulo 9 we get
x 1 2 4 5 7 8
1 1 2 4 5 7 8
2 2 4 8 1 5 7
4 4 8 7 2 1 5
5 5 1 2 7 8 4
7 7 5 1 8 4 2
8 8 7 5 4 2 1
how do i proceed from here
An isomorphic group is a mathematical concept in group theory where two groups have the same structure, even though they may appear different.
To determine if two groups are isomorphic, you can check if there exists a bijective function between the two groups that preserves the group operation. In simpler terms, this means that the function must map elements from one group to the other in a way that maintains the group structure.
Z2 X Z3 is an example of an isomorphic group, where Z2 represents the group of integers modulo 2 and Z3 represents the group of integers modulo 3. When these two groups are combined using the direct product operation, the resulting group is isomorphic to the group G = (1,2,4,8,10).
G = (1,2,4,8,10) is an example of an isomorphic group, where the elements of the group are represented by the numbers 1, 2, 4, 8, and 10. This group is isomorphic to Z2 X Z3, meaning they have the same group structure.
One real-life application of isomorphic groups is in cryptography, where isomorphisms are used to create secure encryption algorithms. By mapping elements from one group to another, it becomes difficult for hackers to decipher the encrypted information, providing a level of security for sensitive data.