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In this dissertation we completely determine the maximal subgroups of the following finite simple groups: (i) POgX?) and 3D^q) for all prime powers q (ii) 2G2(32m+1) for all integers m (iii) G2(<7) for all odd prime powers q. Moreover, if Go is one of the groups appearing in (i), (ii) or (iii), then we also determine the maximal subgroups of all groups G satisfying: GO

Techniques are available today to attenuate the output sound of the trumpet. All of these techniques involve using passive mutes. Due to the limitations in the sound one can obtain with passive mutes, another solution, using active noise control, is proposed to predictably attenuate the output sound of the trumpet. With the new system, it is theorized any desired output sound can be obtained. Within this thesis a model of the trumpet physics is derived and an investigation of the implementation of two analog feedback controllers and two digital LMS controllers is performed. The model of the trumpet mechanics is studied to understand the trumpet system before applying the control systems. Analysis is performed on the type and the location of the acoustic control actuator and the error sensor to be used. With the chosen actuator and sensor, the two types of controllers are designed and realized. The farfield spectrum of the trumpet's response to a single note is analyzed for each controller and the resulting attenuations compared. The model of the trumpet system is then used to demonstrate the coupling of the trumpet and the player and to show the effects of the controllers on the behavior of the player's embouchure. With the inclusion of the controllers in the trumpet system, the farfield spectrum was successfully attenuated at two harmonics of the tone passed through the trumpet. Testing was not performed with an actual trumpet player due to the high sound pressure levels (160 dB SPL) required from the control actuator. From a derived model of the control actuator, specifications for an acoustic driver capable of delivering the high sound pressure level were calculated. Design and fabrication of the proposed actuator will be completed during future work.