Question

Q. Is H=<Q', *> is a subgroup of G=<R, +>

Answer 1

Answer on Question #58983 – Math – Abstract Algebra

Question

Is $H = \langle Q', * \rangle$ is a subgroup of $G = \langle R, + \rangle$?

Solution

If $H = \langle Q', * \rangle$ is a subgroup of $G = \langle R, + \rangle$, then $\langle Q', * \rangle$ must be the group on the operation $+$ specified in $G = \langle R, + \rangle$.

1. Closure: $\forall g_1, g_2 \in H = \langle Q', * \rangle, g_1 + g_2 \in H = \langle Q', * \rangle$.

2. Identity element: $H = \langle Q', * \rangle$ contains 0.

$\forall g_1, g_2 \in H = \langle Q', * \rangle, g_1 + (-g_1) = 0 \in H = \langle Q', * \rangle$.

3. Inverse element: $\forall g_1 \in H = \langle Q', * \rangle, (g_1)^{-1} = (-g_1) \in H = \langle Q', * \rangle$.

4. Associativity: $\forall g_1, g_2, g_3 \in H = \langle Q', * \rangle, g_1 + (g_2 + g_3) = (g_1 + g_2) + g_3$ holds.

Hence the four properties of the subgroup criteria all hold, so $H = \langle Q', * \rangle$ is a subgroup of $G = \langle R, + \rangle$.

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