# Get A Characterization of Alternating Groups by the Set of PDF

By Chen G.

Read or Download A Characterization of Alternating Groups by the Set of Orders of Maximal Abelian Subgroups PDF

Best symmetry and group books

Get NOVELL GroupWise 7 Administrator Solutions Guide PDF

Novell GroupWise 7 Administrator's consultant is the authoritative consultant for effectively administrating and keeping the most recent free up of Novell's verbal exchange and collaboration resolution. writer Tay Kratzer, a Novell top class Service-Primary aid Engineer, provides you with insider tips about management recommendations, confirmed details on how you can paintings with GroupWise 7, and strategies for troubleshooting this most recent unlock of GroupWise now not availalbe within the ordinary GroupWise 7 documentation.

Additional info for A Characterization of Alternating Groups by the Set of Orders of Maximal Abelian Subgroups

Example text

14) are actually in the Lie algebra of the Euclidean group. A simple computation shows that for n≥1  n   y1  ..   Yn Y n−1 y   Y .  ,   =    yn  0 ··· 0 0 ··· 0 where y is the column vector with entries y1 , · · · , yn . 14), then etX is of the form   ∗  ..   etY .  etX =  .  ∗  0 ··· 0 1 40 3. LIE ALGEBRAS AND THE EXPONENTIAL MAPPING Now, we have already established that etY is in O(n) for all t if and only if Y = −Y . 14) with Y satisfying Y tr = −Y . A similar argument shows that the Lie algebra of P(n; 1) is the space of all (n + 2) × (n + 2) real matrices of the form   y1  ..

In our new notation, we may say Ad = ad By the defining property of Ad, we have the following identity: For all X ∈ g, Ad(eX ) = eadX . 17) are linear operators on the Lie algebra g. This is an important relation, which can also be verified directly, by expanding out both sides. 1. Structure Constants. Let g be a finite-dimensional real or complex Lie algebra, and let X1 , · · · , Xn be a basis for g (as a vector space). Then for each i, j, [Xi , Xj ] can be written uniquely in the form n [Xi , Xj ] = cijk Xk .

Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 8. Proof. ) Cm ≤ const. m2 . Since e e domain of the logarithm for all sufficiently large m. 7 Cm ≤ const. the logarithm gives X X Y + + Cm m m X m + Y m + Cm 2 ≤ const. m2 . Exponentiating Y X + + Cm + Em m m Y e m e m = exp and X Y m emem = exp (X + Y + mCm + mEm ) . Since both Cm and Em are of order exponential) lim m→∞ X Y em em 1 m2 , we have (using the continuity of the m = exp (X + Y ) which is the Lie product formula. 10. Let X be an n × n real or complex matrix.