Question

exercise 13 from alperin book, groups and representations.

thanks in advance for explanstions.

extension if it is semalect pioa 12. (cont.) Let Q be the quaternion group of order 8. (We can consider Q as the set仕1, ±i, ±j, ±k} with multiplication given by the rules i2 = j2 = k2 1 and ij = K--ji.) Show that Q can be realized as a non-trivial extension in four ways-thrice as an extension of 4 by Z2, and once as an extension of Z2 by Z2 x Z2-but that none of these extensions is split. (In other words, Q cannot be written non-trivially as a semidirect product.)

Answer 1

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