Find a subgroup of $\left (\mathbb R -\{0\}, \times\right)$ with a finite number of elements, which is not just the trivial subgroup $\{1\}$.
Find a subgroup of $\left(\mathbb R − \{0\}, \times\right)$ with an infinite number of elements, which is not just $\mathbb R − \{0\}$ itself.
Is it possible to find a subgroup of $\left(\mathbb R, + \right)$ with a finite number of elements which is not just the trivial subgroup $\{0\}$?
I've been given these three questions for homework but I'm struggling quiet a bit with them. Thanks to anyone who is able to help.
1) {1,-1}. Identity is 1, $-1^{-1}=-1$
Can prove it is the only one by considering if $b$ is in the group so are all infinite $b^n$.
2) let $b$ be any real number not equal to 1 or -1. Then {$b^n$, *} is an infinite group. 1 is the identity and $b^n*b^m=b^{m+n} $. $(b^n)^{-1}=b^{-n} $.
3) this can't be done. If $b $ in the group then so is $b+b,b+b+b,4b,5b,...nb $ for any possible integer n. This must be in infinite set.