## Cauchy theorem: a good place to review elementary group theory stuff

Cauchy’s theorem in group theory is an excellent place to review fundamental group stuff because (one of) its proof(s) involves the concepts of normal group, centralizer, orbit-stabilizer theorem and conjugacy class equation, etc. On the other hand, it is a crucial step to many important $p$-group results, including Sylow’s theorems.

Theorem (Cauchy):

$G$ a finite group and $p$ a prime. Then if $p\big| |G|$ $\Rightarrow$ $\exists g\in G$ s.t. $g$‘s order is $p$.

Proof:

(i)When $G$ is abelian: do induction on $|G|$. For $|G|=n$: consider $a \in G$, $a\neq e$. If $p\big |\langle a\rangle|$, done; otherwise $p\big| [G:\langle a\rangle]$, by inductive hypothesis $\exists x\langle a\rangle$ of order $p$ in $G/\langle a\rangle$ for some $x\in G$. If $m$ is the order of $x$ in $G$, then $(x\langle a\rangle)^m=\langle a\rangle\Rightarrow p|m$, done.

(ii)For a general $G$: also do induction on $|G|$. for $|G|=n$: $Z(G)$ is abelian. If $p\big| |Z(G)|$, done by (i); otherwise from conjugacy class equation, $\exists \text{Cl}(a)$, where $a\notin Z(G)$, s.t. $p\nmid |\text{Cl}(a)|=[G:C_G(a)]\Rightarrow p\big||C_G(a)|, done by induction.

## Hilbert’s fifth problem and Gleason metrics

The easy version of this question is: when a topological group is a Lie group?

Hilbert’s fifth problem asks to clarify the extent that the assumption on a differentiable or smooth structure is actually needed in the theory of Lie groups and their actions. While this question is not precisely formulated and is thus open to some interpretation, the following result of Gleason and Montgomery-Zippin answers at least one aspect of this question:

Theorem 1 (Hilbert’s fifth problem) Let $latex {G}&fg=000000$ be a topological group which is locally Euclidean (i.e. it is a topological manifold). Then $latex {G}&fg=000000$ is isomorphic to a Lie group.

Theorem 1 can be viewed as an application of the more general structural theory of locally compact groups. In particular, Theorem 1 can be deduced from the following structural theorem of Gleason and Yamabe:

Theorem 2 (Gleason-Yamabe theorem) Let $latex {G}&fg=000000$ be a locally compact group, and let $latex {U}&fg=000000$ be an open neighbourhood of the identity in \$latex…

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## Notes on Group Extension

Here is notes on group extension I wrote, both for the modern algebra class and for my research:

220-notes

The group extension section follows closely section 9.1 of An Introduction to Homological Algebra by Jeseph Rotman.

By the time you see it, I may have added a correspondence between group extension and the PSG used in classifying U(1) Z2 spin liquids.

## A one-sentence account for IQHE

• While the Fermi level crosses a region of localized states, the direct conductivity must vanish whereas the Hall conductivity cannot change. (Not obvious)
• Changing the filling factor therefore will force the Fermi level to change continuously within this region of localized states while the Hall conductance will stay constant. This is the main mechanism leading to the existence of plateaux.

Localization is really the KEY to IQHE.

Reference: The Non-Commutative Geometry of the Quantum Hall Effect, https://arxiv.org/pdf/cond-mat/9411052v1.pdf .