@mina_world: "Hello teacher~
Are the groups Z_8 x Z_10 x Z_24 and Z_4 x Z_12 x Z_40
isomorphic ?
-
Z_8 x Z_10 x Z_24 ~ Z_8 x (Z_2 x Z_5) x (Z_3 x Z_8)
Z_4 x Z_12 x Z_40 ~ Z_4 x (Z_3 x Z_4) x (Z_5 x Z_8)
so,
Z_8 x Z_10 x Z_24 ~ Z_8 x Z_2 x (Z_3 x Z_5 x Z_8)
Z_4 x Z_12 x Z_40 ~ Z_4 x Z_4 x (Z_3 x Z_5 x Z_8)
Of course Z_8 x Z_2 is not isomorphic to Z_4 x Z_4.
Because, Z_8 x Z_2 have an element of order 8.
But Z_4 x Z_4 have an element of maximum order 4.
And then, Can I say that
"Z_8 x Z_10 x Z_24 and Z_4 x Z_12 x Z_40 are not isomorphic." ?
Why ? "
    @mina_world: ""Arturo Magidin" <email_address> wrote in message 
news:g1l9ph$1i4l$<email_address>
<quote>
Yes, it's really fundamental theorem.
The direct product is unique except for possible rearrangement
of the factors with the prime powers.
Upgrade is not easy.
Now, upgraded Mina by Arturo. "
    @rubrum: "<email_address>,
<quote>
The pod people have taken over mina_world;
unless it is the Democratic People's Republic of Korea.
<<url>
Michael Press"
    @rubrum: "<quote>
I am sorry, but this is not clear to me.
The definition of invariant factor decomposition of a 
finite abelian group is
    Z_a x Z_b x Z_c x ... x Z_k
    where a | b | c | ... | k.  
Yes?
Z_8 x Z_10 x Z_24 ~ Z_8 x (Z_2 x Z_5) x (Z_8 x Z_3)
                  ~ Z_2 x Z_8 x Z_8 x Z_3 x Z_5
                  ~ Z_2 x Z_8 x Z_120              < Invariant factor decomposition?
Z_4 x Z_12 x Z_40 ~ Z_4 x Z_4 x Z_3 x Z_8 x Z_5
                  ~ Z_4 x Z_4 x Z_120              < Invariant factor decomposition?
I am new to this.
Michael Press"
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