- #1
jeff1evesque
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Theorem:
A subset H of a group G is a subgroup of G if and only if:
1. H is closed under binary operation of G,
2. The identity element e of G is in H,
3. For all a [tex]\in[/tex] H it is true that [tex]a^{-1}[/tex] [tex]\in[/tex] H also.
Proof:
The fact that if H [tex]\leq[/tex] G then Conditions 1, 2, and 3 must hold follows at one from the definition of a subgroup.
Conversely, suppose H is a subset of a group G such that Conditions 1, 2, and 3 hold. By 2 we have the identity group axiom ([tex]e * x = x *e = x[/tex]) fulfilled. Also, the inverse group axiom ([tex]a * a^{-1} = a^{-1} * a = e[/tex])is fulfilled by Condition 3 (from above). It remains to check the associative axiom. But surely for all a, b, c [tex]\in[/tex] H it is true that [tex](ab)c = a(bc)[/tex] in H, for we may actually view this as an equation in G, where the associative law holds. Hence H [tex]\leq[/tex] G.
Question:
Can someone explain to me the statement above:
The statement in this proof assumes that the associativity is somewhat "obvious", can someone explain the nature of this concept?
Thanks,JL
A subset H of a group G is a subgroup of G if and only if:
1. H is closed under binary operation of G,
2. The identity element e of G is in H,
3. For all a [tex]\in[/tex] H it is true that [tex]a^{-1}[/tex] [tex]\in[/tex] H also.
Proof:
The fact that if H [tex]\leq[/tex] G then Conditions 1, 2, and 3 must hold follows at one from the definition of a subgroup.
Conversely, suppose H is a subset of a group G such that Conditions 1, 2, and 3 hold. By 2 we have the identity group axiom ([tex]e * x = x *e = x[/tex]) fulfilled. Also, the inverse group axiom ([tex]a * a^{-1} = a^{-1} * a = e[/tex])is fulfilled by Condition 3 (from above). It remains to check the associative axiom. But surely for all a, b, c [tex]\in[/tex] H it is true that [tex](ab)c = a(bc)[/tex] in H, for we may actually view this as an equation in G, where the associative law holds. Hence H [tex]\leq[/tex] G.
Question:
Can someone explain to me the statement above:
It remains to check the associative axiom. But surely for all a, b, c [tex]\in[/tex] H it is true that [tex](ab)c = a(bc)[/tex] in H, for we may actually view this as an equation in G, where the associative law holds.
The statement in this proof assumes that the associativity is somewhat "obvious", can someone explain the nature of this concept?
Thanks,JL