Thread: Finite group with identity subgroup.

1. 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?

2. Do you agree that, by definition, a group $\displaystyle G$ is cyclic iff there is some $\displaystyle a$ in $\displaystyle G$ such that $\displaystyle <a>=G$.

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.