Question

Q. Prove that <z,+> is a group, H=<5z,+> is a subgroup of G=<Z,+>

Answer 1

Answer on Question #58981 – Math – Abstract Algebra

Question

Prove that

i) $< \mathbb{Z}, + >$ is a group,

ii) $\mathbb{H} = < 5\mathbb{Z}, + >$ is a subgroup of $\mathbb{G} = < \mathbb{Z}, + >$

Solution

i) Given any integer numbers $a \in \mathbb{Z}, b \in \mathbb{Z}$ , we have to check group axioms w.r.t addition:

1. Closure:

We have $a + b = c$ , where $c \in \mathbb{Z}$ is some integer. Axiom is satisfied.

2. Identity element:

We have $a + 0 = 0 + a = a$ . Thus, identity element is $e = 0$ . Axiom is satisfied.

3. Inverse element:

We have $a + (-a) = (-a) + a = e$ . Thus, the inverse element is $a^{-1} = -a$ .

Axiom is satisfied.

4. Associativity:

As $a + (b + c) = (a + b) + c$ is obviously satisfied for any integers: $a \in \mathbb{Z}, b \in \mathbb{Z}, c \in \mathbb{Z}$ .

Axiom is satisfied.

Therefore, integer numbers satisfy all group axioms w.r.t addition.

ii) Now let us consider $\mathbb{H} = < 5\mathbb{Z}, + >$ . Since $\mathbb{H} \subset \mathbb{Z}$ , to prove that $\mathbb{H} = < 5\mathbb{Z}, + >$ is a subgroup of $< \mathbb{Z}, + >$ , we have to prove that $< \mathbb{H}, + >$ forms a group, i.e. we must check group axioms.

1. Closure:

$$a \in \mathbb {H}, \quad a = 5 n, n = 0, \pm 1, \pm 2, \dots$$ $$b \in \mathbb {H}, \quad a = 5 m, m = 0, \pm 1, \pm 2, \dots$$

Then

$$a + b = 5 n + 5 m = 5 (n + m) \in \mathbb {H}$$

Closure axiom is satisfied.

2. Identity element:

We have $5n + 0 = 0 + 5n = 5n$ . Thus, the identity element is $e = 0 \in \mathbb{H}$ . Axiom is satisfied.

3. Inverse element:

We have $5n + (-5n) = (-5n) + 5n = e$ . Thus, the inverse element is $(5n)^{-1} = -5n \in \mathbb{H}$ .

Axiom is satisfied.

4. Associativity:

As $5n + (5m + 5k) = (5n + 5m) + 5k$, where $c = 5k$, $k = 0, \pm 1, \pm 2, \ldots$, associativity is obviously satisfied for any integers $n \in \mathbb{Z}, m \in \mathbb{Z}, k \in \mathbb{Z}$. Axiom is satisfied.

Therefore, we proved that $\mathbb{H} = < 5\mathbb{Z}, + >$ is a subgroup of $\mathbb{G} = < \mathbb{Z}, + >$.

Answer: $< \mathbb{Z}, + >$ is a group, $\mathbb{H} = < 5\mathbb{Z}, + >$ is a subgroup of $\mathbb{G} = < \mathbb{Z}, + >$.

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