find all subgroups of : a) grup (Z_6 , +) b) grup (Z


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Question Number 157438 by apriadodir last updated on 23/Oct/21
$find all subgroups of : a) grup (Z_6 , +) b) grup (Z_6 −{0}, ×)$
$find all subgroups of :$ $a) grup (Z_{6}, +)$ $b) grup (Z_{6} - {0}, \times)$
	Answered by mindispower last updated on 23/Oct/21
	$if G is a subgroup of E card(G)∣card(E)⇒card(G)∈{1,2,3} 0∈G card(G), since (Z_6_, ,+) is Cycl groupe (Z_6 ,+)=<a> withe a 6=1,a∈F={1,3,5} if a=2, A=<2>,card(A)=3, a=3,A=<3> card(A)=2 a=4,card(A)=3,A=0 card(A)=1 a∈F (z_6 ,+)=<a>=card =6 b) card grup (z_6 −{0},∗) is not a grup (2),(3)∈ but 2∗3=6=0[6]∉$
	$i f G i s a s u b g r o u p o f E$ $c a r d (G) ∣ c a r d (E) \Rightarrow c a r d (G) \in {1, 2, 3}$ $0 \in G c a r d (G),$ $s i n c e (Z_{6_{,}}, +) i s C y c l g r o u p e$ $(Z_{6}, +) =< a > w i t h e a 6 = 1, a \in F = {1, 3, 5}$ $i f a = 2, A =< 2 >, c a r d (A) = 3,$ $a = 3, A =< 3 > c a r d (A) = 2$ $a = 4, c a r d (A) = 3, A = 0 c a r d (A) = 1$ $a \in F (z_{6}, +) =< a >= c a r d = 6$ $b) c a r d g r u p (z_{6} - {0}, ) i s n o t a g r u p$ $(2), (3) \in b u t 2 3 = 6 = 0 [6] \notin$
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