Thread: Which Group is this Group Isomorphic to?

Hello!

Using the external direct product , the group
is isomorphic to one (and only one!) of the following groups (this question reminds me of a game show...)

a)
b)
c)
d)

I am assuming these groups are the well known groups
are the integers modulo n under addittion
is the alternating group (all even permutations of n) under function composition
is the dihedral group of order
and is the set of permutations of n

The question recommends determining which groups it is not isomorphic to and determining the answer by elimination.

I started by the question noting that is cyclic, I believe? so it must be isomorphic to a cyclic group as well...
and also it must have the same number of elements of each order of any group it is isomorphic... but even armed with those facts I am having trouble finding that any of a), b), c) or d) are isomorphic to the given group

Any help appreciated!
Thank you!

Actually, cannot be cyclic. It isn't even abelian! With this in mind, you can remove (a) and (b) from consideration.

To deal with (c) and (d), I would try looking at the orders at some elements in each group. Compare them with your .

Originally Posted by roninpro

Actually, cannot be cyclic. It isn't even abelian! With this in mind, you can remove (a) and (b) from consideration.

To deal with (c) and (d), I would try looking at the orders at some elements in each group. Compare them with your .

Alternatively you can show that . To see this note that and but has no subgroup of order .