If H is a p-group show every element can be multiplied by some power to equal p.

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- Apr 11th 2010, 01:37 PMkimberuOrders of p-groups, normal subgroups
If H is a p-group show every element can be multiplied by some power to equal p.

- Apr 11th 2010, 02:22 PMFancyMouse
a. No need to bring in Cauchy. For any x in G, if x is not the identity, then x has an order p^r for some r>0. What about ?

b. Further hint: prove that and , using the fact that N,L are normal in G

c. Suppose A is the subgroup of N of order 2, B is the subgroup of L of order 3. We know that everything in N and L commutes. In particular, everything in A commutes with everything in B. But A and B are cyclic. What is the subgroup AB look like? (Notice that the "subgroup" AB makes sense because and everything in A commutes with everything in B) - Apr 11th 2010, 02:40 PMmaddas
(b) is so

*intuitive*to me, but the proof your supposed to find is so*opaque*. Can't you use that G is the semidirect product of N and L somehow? - Apr 11th 2010, 03:25 PMkimberu
- Apr 11th 2010, 05:06 PMaliceinwonderland
- Apr 11th 2010, 05:21 PMkimberu
- Apr 11th 2010, 05:26 PMaliceinwonderland