The solvability of some quotient group
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Hello everyone, here's a problem I need help with!
Let G be a group, and let K be a subgroup of G.
If K_i is normal to G, for each i = 1, 2, ..., n,
write K = K_1 (intersect) K_2 (intersect) ... (intersect) K_n.
THE QUESTION: If G / K_i is solvable for each i, show that G / K is solvable.
I'm sorry if it's hard to read, so I've attached a copy of the question as an image, just in case.
Thanks for your time.
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Solution Summary
It is proved that if the quotient of a group by each of the given normal subgroups is solvable, then the quotient of this group by the intersection of these subgroups is also solvable, provided that the number of subgroups is finite.
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