Product of Groups: Understand Max Subgroups & Taking the Product of Groups

[DanielThrice]

I'm having trouble understanding the product of groups and their max normal subgroups. What does it mean to be a max subgroup? How do I take the product of two groups?
How do I do it for something like S₇ X Z₇ ?

 

[Tinyboss]

It's just the cartesian product (like you can make out of any collection of sets), and the group operation on the product is done componentwise, using the operation from each factor group. Surely this definition is in your textbook.

A maximal (normal) subgroup is a (normal) subgroup not properly contained in another proper (normal) subgroup. Note in particular this does not imply that a maximal normal subgroup contains every other normal subgroup.

 

[Beeraa]

This is my first post on this forum, so I hope I don't break any rules and give you too much help for your homework :p

But I remember being very confused when I first bumped into the direct product of groups.

If we start with the basic definition: If A and B are both groups then A x B = {(a,b) | a \in A, b\in B}, so it is the set of all (a,b) where a is in A and b is in B.

then for example to find Z₂ x Z₃:

we know {0,1} is Z₂ and {0,1,2} is Z₃
so Z₂ x Z₃ is the set of all (a,b) where a is in {0,1} and b is in {0,1,2}
therefore Z₂ x Z₃ = {(0,0), (0,1), (0,2), (1,0), (1,1), (1,2)}

obviously this is very clumsy and long, so if you're working with direct products it's useful to note that when Z_n x Z_m, if n and m are coprime then Z_n x Z_m is isomorphic to Z_nm.

So in the example above, Z₂ x Z₃ is isomorphic to Z₆.