2. a. True. Without loss of generality, assume that m is the x-axis, t is the translation to the right by a units, and g is a glide reflection with respect to the x-axis to the right by b units. (Note that a and b may be negative.)
Then gt(x,y)=tg(x,y)=(x+a+b,-y).
b. Let f be a reflection with respect to l and g be a non-trivial rotation at C. Since fg reverses orientation, it is either a reflection or a glide reflection. Hence fg is a reflection iff (fg)2=1 iff (fg)2(C)=C iff f(C)=C iff C is on l. Therfore the answer is: fg is a reflection iff C is on l, and otherwise it is a glide reflection.
3. Note that SO2 and O2 are subgroups of E. We proved that a finite subgroup of E is either cyclic (when it consists of rotations) or dihedral (when it contains reflections). Thus a finite subgroup of SO2 is cyclic. And a finite subgroup of O2 is cyclic or dihedral.
4. The rotational symmetry group of a "thick pentagon" is D5 and of a thick pentagon with black bottom and white top is Z5. The full symmetry group of a thick pentagon with an F on each slice of its bottom and top is D5 and of a thick pentagon with an F on each slice of only its top is Z5.
5. a. Use the results: E+=E0+T and E0+ is isomoprhic to SO2.
b. Use the corresponding results about E.
Bonus. The longest word in capitals was found by
Burcu Baran: CHECKBOOK,
the second longest by Duygu Aruğaslan: DECODE
Please inform me if you find something longer.