Different groups of radio buttons with the same name

Question

I need to create several groups of radio buttons, grouped by a concept (collapsible) but the radio buttons must have the same name so that the form only sends one selected option among all the groups

Group 1
() Radio 1
() Radio 2
() Radio 3
Group 2
() Radio 4
() Radio 5
Group 3
() Radio 6
() Radio 7

I am developing a shipping method of commerce_shipping This form is for choose a shipping service details, therefore I only generate a part of the form of the checkout page in the function that defines the details of the service, using the commerce_shipping API. I test to set the #name property for all radio buttons but I lose the form tree, and I need to keep it so that another function can receive the data. Being inside commerce_shipping I do not have absolute control either.

The question is: What Form API elements should I use?

I tested

$form['group_1']  = array (
  '#type' => 'fieldset',
  '#title' => t('Group 1:'),
  '#container' => TRUE,
  '#collapsible' => TRUE,
  '#collapsed' => FALSE,  
);
$form['group_1']['radios'][] = array (
  '#title' =>  'Radio 1',
  '#type'=>'radio',
  '#return_value' => '1',
);

But ther resulting radio button name is group_1][radios][0] and the next is group_1][radios][1], as the name is different then they are not part of the same selection

Answer 1

Every group is a fieldset

  $form['pickup_group']['term_'.$tid] = array (
    '#type' => 'fieldset',
    '#title' => $pickupWrapper->field_taxonomy->name->value(),
    '#container' => TRUE,
    '#collapsible' => TRUE,
    '#collapsed' => FALSE,  
  );

Every radio

  $form['pickup_group']['term_'.$tid]['shipping_pickup_point'][] = array (
    '#title' =>  drupal_render($renderable_node),
    '#type'=>'radio',
    '#return_value' => $pickupWrapper->getIdentifier(),
    '#parents' => array('commerce_shipping','service_details','shipping_pickup_point'),
  );

I solved adding the array of parents and itself in #parents on every radio, I don't like so much because I don't have control about this parents. My code doesn't generate this parents it comes from the commerce_shipping API. But works...