What does \(H\le G\) mean?

Does a subgroup use a new operation or the parent group's operation \(*\)?

Name one subgroup of \((\mathbb Z,+)\).

What identity element must a subgroup of \((\mathbb Z,+)\) contain?

Show that \(2\mathbb Z\) is closed under addition.

Show that \(2\mathbb Z\) is closed under additive inverses.

Is \(\{1,-1\}\) a subgroup of the non-zero real numbers under multiplication?

Why is \(\{1,2\}\) not a subgroup of the non-zero real numbers under multiplication?

Test whether \(3\mathbb Z\) is a subgroup of \((\mathbb Z,+)\).

Test whether the positive integers are a subgroup of \((\mathbb Z,+)\).

Why does associativity not need a new proof when testing a subgroup \(H\le G\)?

In \(D_4\), explain why \(\{e,r,r^2,r^3\}\) is a subgroup.

Test whether \(\{e,r^2\}\) is a subgroup of the rotations of a square.

Test whether \(\{e,s\}\) is a subgroup of a dihedral group when \(s\) is a reflection.

Show that the intersection of two subgroups \(H\) and \(K\) of \(G\) is a subgroup of \(G\).

Why is the union of two subgroups not always a subgroup? Give an example in \((\mathbb Z,+)\).

Let \(H\) be a non-empty finite subset of a group \(G\) that is closed under products. Explain why \(H\) is a subgroup.

A student tests a subgroup candidate by changing multiplication to addition inside the subset. Diagnose the error.

Prove that every subgroup contains the parent identity.

If \(H\le G\) and \(g\in G\), why is the set \(gHg^{-1}=\{ghg^{-1}:h\in H\}\) a subgroup?