While searching for non-isomorph subgroups of order 2002 I just encountered something, which I want to understand. Obviously I looked for abelian subgroups first and found so we have the groups
Now I want to understand why those two are not isomorph. I know that for two groups it has to hold that . But I don't understand how we can compare Groups written as two products with groups written as three products as above, how does that work? And I think that goes in the same direction: How is it then at the same time that
because . I don't understand the difference to the first comparison.