- - - - - - Group analysis of a cube - - - - - - - - - - | | i.e "cube group" - rotational symmetries of a solid 3-dimensional cube | - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- +-----+ / U /| names of the sides are F front +-----+ | (i.e. colors) B back | F |R+ R right | |/ L left +-----+ T up D down U | L F R B | I'll list the sides like this, "unfolded" D | Let " F " be the move which is a 90 degree clockwise rotation about the Front face, while looking straight at it. Let " F' " be its inverse, counterclockwise rotation, F FR = D R U L = (FUR)(BDL) permutation cycle B UF = same = RU (Note that therefore FR=UF and U = F R F' ) B RB = U R D L = (FDR)(ULB) F F FRU = R U L D = (RL)(FU)(BD) = one of the 180-edge rotations B ======= Here's the output of a computer program I wrote to do this out completely: The 24 distinct elements as moves showing an "unfolded" cube, starting with "I", the identity. ================================================== 1 I : U L F R B D --------------------- 2 F : L D F U B R --------------------- 3 R : F L D R U B --------------------- 4 U : U F R B L D --------------------- 5 B : R U F D B L --------------------- 6 L : B L U R D F --------------------- 7 D : U B L F R D --------------------- 8 F2 : D R F L B U --------------------- 9 R2 : D L B R F U --------------------- 10 U2 : U R B L F D --------------------- 11 FR : F D R U L B --------------------- 12 RB : R F D B U L --------------------- 13 BL : B U R D L F --------------------- 14 LF : L F U B D R --------------------- 15 LB : R B U F D L --------------------- 16 FL : B D L U R F --------------------- 17 RF : L B D F U R --------------------- 18 BR : F U L D R B --------------------- 19 FRU : F R U L D B --------------------- 20 RUF : D B R F L U --------------------- 21 UFR : R D B U F L --------------------- 22 BUR : D F L B R U --------------------- 23 LFU : L U B D F R --------------------- 24 DRF : B R D L U F --------------------- And here's the full group multiplication table using the names given above for the group elements. ==================================================================================================== 1 1 2 3 4 5 6 7 8 9 0 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 Conjugate classes and some properties (angle and axis of roation, n=number of elements, c=cycle length): Identify, n=1, c=1 . 90 deg face, n=6, c=4 . . 180 deg face, n=3, c=2 . . . 120 deg corner, n=8, c=3 . . . . 180 deg edge, n=6, c=2 . . . . . . ..................... .......... .............................. ....................... I F R U B L D F2 R2 U2 FR RB BL LF LB FL RF BR FRU RUF UFR BUR LFU DRF ----------------------------------------------------------------------------------------------- I |I F R U B L D F2 R2 U2 FR RB BL LF LB FL RF BR FRU RUF UFR BUR LFU DRF F |F F2 RF FR I LF FL B LFU UFR RUF R U FRU L BUR DRF D LB BL R2 BR U2 RB R |R FR R2 RB BR I RF FRU L DRF UFR BUR B U D F RUF LFU U2 LB FL LF BL F2 U |U LF FR U2 RB BL I BUR RUF D FRU UFR DRF LFU B L F R BR F2 LB R2 RF FL B |B I RB BL F2 LB BR F UFR LFU U DRF RUF L FRU D R BUR LF FR U2 FL R2 RF L |L FL I LF BL R2 LB DRF R FRU F U LFU BUR RUF UFR D B F2 RF FR RB BR U2 D |D RF BR I LB FL U2 RUF BUR U R B L F UFR DRF LFU FRU FR R2 RB F2 LF BL F2 |F2 B DRF RUF F FRU BUR I U2 R2 BL RF FR LB LF BR RB FL L U LFU D UFR R R2 |R2 UFR L BUR LFU R RUF U2 I F2 FL LF BR RB RF FR LB BL DRF D F U B FRU U2 |U2 LFU FRU D UFR DRF U R2 F2 I BR LB FL RF RB BL LF FR R BUR B RUF F L FR |FR FRU RUF UFR R U F BR BL FL LB R2 RB U2 I LF F2 RF D B L LFU DRF BUR RB |RB U UFR DRF BUR B R LF LB RF U2 FL F2 BL BR I FR R2 LFU FRU D L RUF F BL |BL L U LFU DRF RUF B FL FR BR LF U2 RF R2 F2 LB I RB BUR F FRU UFR R D LF |LF BUR F FRU U LFU L RB RF LB F2 FR U2 BR BL R2 FL I B DRF RUF R D UFR LB |LB D B L RUF UFR FRU RF RB LF I BL R2 FL FR U2 BR F2 F R U DRF BUR LFU FL |FL DRF D F L BUR UFR BL BR FR RF I LF F2 R2 RB U2 LB RUF LFU R B FRU U RF |RF RUF LFU R D F DRF LB LF RB R2 BR I FR FL F2 BL U2 UFR L BUR FRU U B BR |BR R BUR B FRU D LFU FR FL BL RB F2 LB I U2 RF R2 LF U UFR DRF F L RUF FRU |FRU BR F2 LB FR U2 LF R DRF L B RUF UFR D U LFU BUR F I RB BL RF FL R2 RUF |RUF LB BL R2 RF FR F2 D U BUR L LFU R UFR F FRU B DRF FL I LF U2 RB BR UFR |UFR U2 LB FL R2 RB FR LFU B F D L BUR DRF R U FRU RUF RF BR I BL F2 LF BUR |BUR RB FL F2 LF BR R2 U D RUF DRF F FRU B LFU R UFR L BL U2 RF I LB FR LFU |LFU R2 LF BR U2 RF BL UFR F B BUR FRU D R DRF RUF L U RB FL F2 FR I LB DRF |DRF BL U2 RF FL F2 RB L FRU R LFU D F RUF BUR B U UFR R2 LF BR LB FR I | ==================================================================================================== Subgroups: Cyclic (single generator): { I, F2 } { I, R2 } { I, U2 } { I, FRU } { I, RUF } { I, UFR } { I, BUR } { I, LFU } { I, DRF } { I, FR, LB } { I, RB, FL } { I, BL, RF } { I, LF, BR } { I, F, F2, B } { I, R, R2, L } { I, U, U2, D } Commutator subgroup (All g=xyx'y' for some x,y - here turns out to be even number of F,R,U,B,L,D moves: { I, F2, R2, U2, FR, RB, BL, LF, LB, FL, RF, BR } ? name ? {I, F2, R2, U2} Trivial (identify only, entire group): { I } { I, F, R, U, B, L, D, F2, R2, U2, FR, RB, BL, LF, LB, FL, RF, BR, FRU, RUF, UFR, BUR, LFU, DRF } Any others?