Hecke groups $H(\lambda _q)$ are the discrete subgroups of ${\mathrm PSL}(2,\mathbb{R})$ generated by $S(z)=-(z+\lambda _q)^{-1}$ and $T(z)=-\frac{1}{z} $. The commutator subgroup of $H$($\lambda _q)$, denoted by $H^{\prime }(\lambda _q)$, is studied in [2]. It was shown that $H^{\prime }(\lambda _q)$ is a free group of rank $q-1$. Here the extended Hecke groups $\bar{H}(\lambda _q)$, obtained by adjoining $R_1(z)=1/\bar{z}$ to the generators of $H(\lambda _q)$, are considered. The commutator subgroup of $\bar{H}(\lambda _q)$ is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the $H(\lambda _q)$ case, the index of $H^{\prime }(\lambda _q)$ is changed by $q$, in the case of $\bar{H}(\lambda _q)$, this number is either 4 for $q$ odd or 8 for $q$ even.
High nodulating (HN) selections of the cultivars ICC 4948 and ICC 5003 had the highest nodule number and nodule dry mass followed by low nodulating (LN) selections of the same cultivar. Both non-nodulating (NN) selections of cv. ICC 4993 and ICC 4918 did not show any nodule. Using N-difference method the HN selection of cv. 1CC 4948 was able to meet 73 % of its demand of N through biological fixation of N2 [P(fix)], while 27 % of N demand was met by uptake from the soil, whereas its LN selection was able to meet only 54 % of its demand of N through biological fixation of N2. Similarly in cv. ICC 5003 HN and LN selections the P(fix) was 76 and 64 %, respectively. Fast chlorophyll (Chl) fluorescence transient data analysis showed that performance index PI(abs) was 62.0 in cv. ICC 4948 HN selection and 44.5 in its respective LN selections. Corresponding values for cv. ICC 5003 were 32.4 and 28.4. In NN selections of ICC 4993 and ICC 4918 it was 12.6 and 30.7, respectively. Structure function index of the plants SFI(abs) and driving force for photosynthesis (DF) were highest in the HN selections followed by LN selections and lowest in the NN selections. The total uptake of N by chickpea plants was significantly and positively correlated with the density of reaction centres ABS/CS0, TR0/CS0, and DI0/CSM, whereas total N uptake by chickpea seeds was significantly positively correlated with N and TR0/CS0. The percentage of P(fix) was highly significantly positively correlated with N, the so-called turnover number which indicates how many times QA has been reduced in the time span from 0 to tFmax and TR0/CS0. Fast Chl a fluorescence measurement can be used as a model system to assess the N fixation ability in chickpea. and S. S. Dudeja, P. Chaudhary.