A four-bladed marine propeller has its blades pitched (twisted) all in the same sense

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

A four-bladed marine propeller has its blades pitched (twisted) all in the same sense, so no mirror plane, rotation axis other than the shaft, or point of inversion maps it onto itself: the only symmetry operations that leave it looking unchanged are rotations about the shaft axis. Taking that axis as zz, its symmetry operations are the four rotations EE, C4C_4 (90∘90^\circ), C2≡C42C_2\equiv C_4^2 (180∘180^\circ) and C43C_4^3 (270∘270^\circ) about zz, represented by the matrices E=(100010001), C4=(0−10100001),E=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix},\ C_4=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&1\end{pmatrix}, C2=(−1000−10001), C43=(010−100001)C_2=\begin{pmatrix}-1&0&0\\0&-1&0\\0&0&1\end{pmatrix},\ C_4^3=\begin{pmatrix}0&1&0\\-1&0&0\\0&0&1\end{pmatrix} ((a)) Construct the 4×44\times4 multiplication table of {E,C4,C2,C43}\{E, C_4, C_2, C_4^3\}, writing ghgh for "hh first, then gg". ((b)) Confirm the four group axioms from the table and state whether the group is abelian. ((c)) Using Lagrange's theorem, list every possible order of a subgroup, then find how many subgroups of order 22 actually exist. Compare this with a group of the same order built instead from three mutually perpendicular 180∘180^\circ rotations, in which every non-identity element has order 22, and explain the difference. ((d)) List the left cosets of the order-22 subgroup found in part (c).

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