Let $F_0$ be the maximal real subfield of $mathbb{Q}({zeta}_q+{zeta}_q^{-1})$ and $F_{infty}={cup}_{n{geq}0}F_n$ be its basic $mathbb{Z}_p$-extension. Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $F_n$. The aim of this paper is to examine the injectivity of the natural $mapA_n{ ightarrow}A_m$ induced by the inclusion $F_n{ ightarrow}F_m$ when $m>n{geq}0$. By using cyclotomic units of $F_n$ and by applying cohomology theory, one gets the following result: If $p$ does not divide the order of $A_1$, then $A_n{ ightarrow}A_m$ is injective for all $m>n{geq}0$.