Question

7. Let G be a group of order 30. (a) Show that a 3-Sylow subgroup or a 5-Sylow subgroup of G must be normal in G. (b) From part (a) show that every 3-Sylow subgroup and every 5-Sylow subgroup of G must be normal in G. (c) Show that G has a normal subgroup of order 15.

          7. Let G be a group of order 30.
(a) Show that a 3-Sylow subgroup or a 5-Sylow subgroup of G must be normal in G.
(b) From part (a) show that every 3-Sylow subgroup and every 5-Sylow subgroup of G must be normal in G.
(c) Show that G has a normal subgroup of order 15.

7. Let G be a group of order 30.
(a) Show that a 3-Sylow subgroup or a 5-Sylow subgroup of G must be normal in G.
(b) From part (a) show that every 3-Sylow subgroup and every 5-Sylow subgroup of G must be normal in G.
(c) Show that G has a normal subgroup of order 15.

Added by Nerea Q.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

06/24/2024

Step 1

So, $n_3$ divides 10 and $n_3 \equiv 1 \pmod{3}$. The possible values for $n_3$ are 1 and 10. Similarly, the number of 5-Sylow subgroups, $n_5$, must divide the order of the group and be congruent to 1 modulo 5. So, $n_5$ divides 6 and $n_5 \equiv 1 \pmod{5}$. The Show more…

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Let G be a group of order 30. (a) Show that a 3-Sylow subgroup or a 5-Sylow subgroup of G must be normal in G. (b) From part (a) show that every 3-Sylow subgroup and every 5-Sylow subgroup of G must be normal in G. (c) Show that G has a normal subgroup of order 15.