Fine Structure of Class Groups ${\text{cl}}^{(p)}\text{Q}({\text{z}}_n)$ and the Kervaire--Murthy Conjectures II

There is an Mayer-Vietoris exact sequence involving the Picard group of the integer group ring $\mathbb{Z}{C}_{p^n}$ where $C_{p^n}$ is the cyclic group of order $p^n$ and ${\zeta }_{n-1}$ is a primitive $p^n$-th root of unity. The “unknown” part of the sequence is a group. $V_n$. $V_n$ splits as $V_n\cong V_n^+\oplus V_n^-$and$V_n^-$isexplicitlyknown\mathrm{.}$V_n^+$ is a quotient of an in some sense simpler group ${\mathcal{V}}_n$. In 1977 Kervaire and Murthy conjectured that for semi-regular primes $p$, $V_n^+ \cong $\mathcal{V}$n^+ \cong \cl^{(p)}(\Q (\zeta{n-1}))\cong ($\mathbb{Z}$/p^n $\mathbb{Z}$)^{r(p)}$,where$r(p)$istheindexofregularityof$p$\mathrm{.}Underanextraconditionontheprime$p$,Ullomcalculated$V_n^+$in1978intermsoftheIwasawainvariant$\lambda$as$V_n^+ \cong ($\mathbb{Z}$/p^n $\mathbb{Z}$)^{r(p)}\oplus ($\mathbb{Z}$/p^{n-1} $\mathbb{Z}$)^{\lambda-r(p)}$. In the previous paper we proved that for all semi-regular primes, ${\mathcal{V}}_n^{+}\cong {\text{cl}}^{(p)}(\text{Q}({\zeta }_{n-1}))$ and that these groups are isomorphic to $$(\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}{)}^{r_0}\oplus (\mathbb{Z}\mathrm{/}{p}^{n-1}\mathbb{Z}{)}^{r_1-r_0}\oplus \text{hdots}\oplus (\mathbb{Z}\mathrm{/}p\mathbb{Z}{)}^{r_{n-1}-r_{n-2}}$$ for a certain sequence $\{r_k\}$ (where $r_0=r(p)$). Under Ulloms extra condition it was proved that $$V_n^{+}\cong {\mathcal{V}}_n^{+}\cong {\text{cl}}^{(p)}(\text{Q}({\text{z}}_{n-1}))\cong (\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}{)}^{r(p)}\oplus (\mathbb{Z}\mathrm{/}{p}^{n-1}\mathbb{Z}{)}^{\lambda -r(p)}\mathrm{.}$$ In the present paper we prove that Ullom’s extra condition is valid for all semi-regular primes and it is hence shown that the above result holds for all semi-regular primes.