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Section Day 3
This is an outline of the topics we covered in the third day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.
Handout Tuesday 5/26
Objectives: Advanced Learning Outcomes
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
Determine when powers of a group element are equal.
Determine when cyclic subgroups of a cyclic group are equal and their cardinality.
State the following mathematical result: Theorem 4.3 (Fundamental Theorem of Cyclic Groups)
Determine the subgroups of a given order in a cyclic group.
Algebraist of the Day.
Emmy Noether , 1882-1935
Pioneer in commutative algebra, esp. objects satisfying the
ascending chain condition , which are now called
Noetherian
Proved Noether’s Theorem about symmetries & conservation laws in physics.
Worked at at Erlangen without pay for 7 years; overcame resistance to be allowed to teach at Göttingen (4 years under Hilbert’s name)
Proposition 26 . \(K_4\) is the Smallest Non-Cyclic Group.
All groups of order at most
\(3\) are cyclic.
Proof.
Theorem 27 . Equality of Powers of Group Elements.
Let
\(G\) be a group and
\(a\in G\text{.}\) If
\(|a|=\infty\text{,}\) then
\(a^i=a^j\) iff
\(i=j\text{.}\) If
\(|a|=n\text{,}\) then
\(\langle a\rangle =\{e,a,a^2,\dots,a^{n-1}\}\) and
\(a^i=a^j\) iff
\(n\mid (i-j)\text{.}\)
Proof of 27.
Theorem 28 . Fundamental Theorem of Cyclic Groups.
Every subgroup of a cyclic group is cyclic. Moreover, if \(|\subgroup{a}|=n\text{,}\) then
the order of any subgroup of
\(\subgroup{a}\) divides
\(n\)
for each positive divisor
\(k\) of
\(n\text{,}\) \(\subgroup{a}\) has exactly one subgroup of order
\(k\text{,}\) namely
\(\subgroup{a^{n/k}}\text{.}\)
Example 29 . Application of the Fund. Thm to \(\Z_{20}\) .
Proof of Fundamental Theorem of Cyclic Groups.
