Does a number have a primitive root if and only if φ(n)=λ(n)?

Question

From the Wikipedia article on root primitives:

In particular, for a to be a primitive root modulo n, φ(n) has to be the smallest power of a which is congruent to 1 modulo n.

Am I correct if I say a number has a primitive root if and only if $φ(n)=λ(n)$ (that is, Euler's Totient Function and Carmichael's Function)?

Answer 1

If the exponent of a finite abelian group equals the order of the group, then the group must be cyclic. The other direction is a matter of checking that $\varphi(n)=\lambda(n)$ for $n = 2,4,p^k,2p^k$ which is true.