Finite group with identity subgroup.

• Feb 18th 2010, 05:25 AM
chipette
Finite group with identity subgroup.
If you have a finite group, say G with no subgroups apart from {1G } and G. How would you go about showing that G is cyclic and that the number of elements in G is either 1 or prime?

I know that the order of this group is 1,but how would you show that this makes it cyclic, by Lagrange's theorem?
• Feb 18th 2010, 09:17 AM
clic-clac
Do you agree that, by definition, a group \$\displaystyle G\$ is cyclic iff there is some \$\displaystyle a\$ in \$\displaystyle G\$ such that \$\displaystyle <a>=G\$.

Quote:

I know that the order of this group is 1
Since the order of a finite group is its cardinality, the one and only group of order 1 is the trivial group \$\displaystyle G=\{1_G\}.\$ But there are non trivial groups with no subgroup other than \$\displaystyle \{1\}\$ and themselves (i.e. with no non-trivial subgroup).

So, consider a finite group \$\displaystyle G\neq\{1_G\}\$ with no non-trivial subgroup, let's say \$\displaystyle G\$ has order \$\displaystyle n\geq 2\$. You may know that if a prime \$\displaystyle p\$ divides \$\displaystyle n,\$ then \$\displaystyle G\$ has an element of order \$\displaystyle p.\$
Assume \$\displaystyle n\$ is not a prime, and try to obtain a contradiction by proving \$\displaystyle G\$ has a non-trivial subgroup.

So far you'll have that a finite group with no non-trivial subgroup has order 1 or a prime. If it is 1, do you agree that \$\displaystyle G=\{1_G\}=<1_G>\$ therefore \$\displaystyle G\$ is cyclic, because its only element is a generator. If it is not 1, take an element of order strictly greater than \$\displaystyle 1\$ (justify its existence) and consider the subgroup it spans. Because of the hypothesis, since it is different from \$\displaystyle \{1_G\},\$ it must be the whole group, hence \$\displaystyle G\$ is cyclic.

Note that we did not to use that order (when different from 1) is prime; that being said a similar argument shows that any group with order a prime has no non-trivial subgroup (and is cyclic).

Conclusion: A group has no non-trivial subgroup iff its order is 1 or a prime.