Und dann erst kommt der "Ab -ge - sa. ng\' da. /3 der nidlt kurz und nicht zu la. ng, From "Die Meistersinger von Nürnberg", Richard Wagner This final volume is concerned with some of the developments of the subject in the 1960's. In attempting to determine the simple groups, the first step was to settle the conjecture of Burnside that groups of odd order are soluble. The proof that this conjecture was correct is much too long and complicated for presentation in this text, but a number of ideas in the early stages of it led to a local theory of finite groups, so me aspects of which are discussed in Chapter X. Much of this discussion is a con tinuation of the theory of the transfer (see Chapter IV), but we also introduce the generalized Fitting subgroup, which played a basic role in characterization theorems, that is, in descriptions of specific groups in terms of group-theoretical properties alone. One of the earliest and most important such characterizations was given for Zassenhaus groups; this is presented in Chapter XI. Characterizations in terms of the centralizer of an involution are of particular importance in view of the theorem of Brauer and Fowler. In Chapter XII, one such theorem is given, in which the Mathieu group 9J'l1l and PSL(3, 3) are characterized

University Of Chicago Department Of Mathematics.

A fundamental life-history trade-off occurs between the size and number of offspring that a female produces. Traditionally, biologists have assumed that there is a species-specific optimal egg size, the value of which can fluctuate with changing environmental parameters. However, in unpredictable environments a bet-hedging strategy resulting in variable offspring sizes may be favored. The seastar Asterias forbesi can produce viable eggs that vary more than three-fold in volume within a single clutch (141μm - 212μm diameter). Compared to 12 other echinoderm species with similar modes of development (planktotrophic), A. forbesi represents an organism with unusual intra-clutch variation in egg size. In addition, the larvae derived from these eggs have frequently been observed to produce clones. To test for maternal effects on cloning frequency and larval development I reared cultures of large (190μm mean diameter) or small (140μm mean diameter) sibling embryos. Previous studies have shown that exogenous cues can alter the frequency of cloning, but it is unclear whether endogenous reserves might also influence cloning. My results suggest that despite an initial disadvantage in energy reserves, small larvae produced clones at frequencies similar to their larger siblings. Since little is known about the links between maternal investment and juvenile quality in seastars, I continued to follow these larvae and examined the effect of maternal investment on time to and size at metamorphosis. Small larvae took about two additional days to reach metamorphosis compared to large larvae, which was a 6.3% increase in developmental time. Size at metamorphosis did not appear to be affected by maternal investment and varied greatly within size classes. To further examine the costs of delayed metamorphosis, I set up an additional experiment, which examined the effect of time to settlement on post-metamorphic survivability in juveniles. My results indicate that size at settlement, rather than time to settlement, is correlated with survivability.