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PROBLEM: Let ##G## be a dihedral group of order 30. Determine ##O_2(G),O_3(G),O_5(G), E(G),F(G)## and ##R(G).##

Where ##O_p(G)## is the subgroup generated by all subnormal

*p*-subgroups of ##G##; ##E(G)## is the layer subgroup ##G##; ##F(G)## is the fitting subgroup of ##G## and ##R(G) := E(G)F(G)## is the radical subgroup of ##G.##

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Let's focus on determining ##O_2(G),O_3(G),O_5(G)## first, and here are what I got so far:

(1) Suppose that the number of ##Syl_p(G)## in a group is denoted by ##n_p##, and the order ##|Syl_p(G)| := o_p.##

(2) We observe that ##|G| = 30 = 2 \cdot 3 \cdot 5, \ ## all of which are prime, and ##(2, 15) = 1,## therefore if we take ##p = 2## then by Sylow's Theorem ##n_2 \equiv 1 (mode \ 2)## and ##n_2 \mid 15.## Therefore ##n_2 = \{1, 3, 5, 15 \}.## The theorem further implies that ##o_2 = 2.##

(3) If we take ##p = 3##, we observe that ##(3, 10) = 1## therefore by the same Sylow's Theorem we get ##n_3 = \{ 1, 10 \}## and ##o_3 = 3.##

(4) Again if we take ##p = 5##, we observe that ##(5, 6) = 1## therefore by the same Sylow's Theorem we get ##n_5 = \{ 1, 6 \}## and ##o_6 = 6.##

(5) Observe that ##Syl_2(G), Syl_3(G)## and ##Syl_5(G)## are cyclic since their orders are prime, therefore their elements do not overlap, meaning that their intersection consists only of ##\{ e \}##, the neutral element.

(6) Recall that ##Syl_2(G), Syl_3(G)## and ##Syl_5(G)## have to co-exist, and ##|G| = 30,## therefore the only possible combination will be ##n_2 = \{1, 3, 5\}, n_3 = \{1\},## and ##n_5 = \{1\}.## Otherwise the total elements ##|G| > 30.##

(7) ...

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Unfortunately I don't know what is next after line (6), especially I don't know how to incorporate the fact that they are sub-normal and the group is dihedral. Nor do I know if I have been on the right direction. I would therefore love to get help from you. The class note does not offers any hint in solving the problem except a parade of lemmas, corollaries and theorems, one of which I believe is relevant to solving this problem:

Lemma: Let ##p## be a prime, then (i) ##O_p(G)## is the uniquely determined largest normal

*p*-subgroup of ##G##; (ii) ##O_p(G)## is the intersection of all Sylow

*p*-subgroups of ##G.##

Thank you very much for your time and help.