a touch of group theory and throwback to abstract algebra

the story: a little bit of context before the mathematics (feel free to skip)

in my third year i couldn't decide my final module: groups and symmetry or game theory. i went back and forth, changed it a few times before the spring term, probably driving the course officer at the time insane. when the new term rolled around my timetable showed both of the lectures and i had access to both of the moodle pages. i was tagged in a group email the professor sent just before our first lecture. it was timetabled on a monday morning at 9am so he was giving us a warm welcome and a gentle nudge to please not skip his lecture.

i missed the first lecture and in the second one he asked if i was even enrolled on the course. when i said i wasn't sure he replied that i was welcome to audit all the lectures. i eventually figured out i was only officially enrolled in the game theory course, which was a tactical move since the module was easier and i needed to boost my average badly. the professor's offer still stood though. i was welcome to audit the groups and symmetry class, i was even welcome to do a midterm and get it marked, hand plastered with a bunch of "?" and "attempt" in red ink.

today we will go over two of the three questions on that midterm in detail, zooming in and out of specific keywords while answering. it's not enough to just know pure mathematics, you have to be precise. it reminds me of science in school, having to memorise the mark scheme above anything. i always hated science.

objectives: accurately define cyclic group, group theory axioms,

1: a) state the definition of a cyclic group. show that all cyclic groups are abelian

a cyclic group is a group in which there exists a generator a such that for a specific x^n = a

LATEX TEST

this is a test of inline LaTeX: \( a^2 + b^2 = c^2 \)

and here is a displayed equation:

\[ \frac{d}{dx} \left( \int_{0}^{x} f(t)\,dt \right) = f(x) \]