If you try to use RadioButtons as usual in MVVM, it will not work as you expect.
For example, here is a very simple UI using two RadioButtons.

A very simple UI using two RadioButtons.


My first attempt is to use this View:
...
<GroupBox Header="I love">
    <StackPanel>
        <RadioButton IsChecked="{Binding IsPeaceLover}">Peace</RadioButton>
        <RadioButton>Earth</RadioButton>
    </StackPanel>
</GroupBox>
...
and this ViewModel:
class UserViewModel : ViewModelBase
{
    bool isPeaceLover = true;

    public bool IsPeaceLover
    {
        get { return this.isPeaceLover; }
        set
        {
            if (this.isPeaceLover == value)
                return;

            this.isPeaceLover = value;
            OnPropertyChanged("IsPeaceLover");
        }
    }
    ...
}
But as you may know, it doesn't work. There are many solutions to avoid this problem.
Anyway, I've created a new pattern that is more appropriate to MVVM (not using Code-Behind).
Try this View:
...
<GroupBox Header="I love">
    <StackPanel>
        <RadioButton IsChecked="{Binding IsPeaceLover}">Peace</RadioButton>
        <RadioButton IsChecked="{Binding IsEarthLover}">Earth</RadioButton>
    </StackPanel>
</GroupBox>
...
and this ViewModel:
class UserViewModel : ViewModelBase
{
    bool isPeaceLover = true;

    public bool IsPeaceLover
    {
        get { return this.isPeaceLover; }
        set
        {
            if (this.isPeaceLover == value)
                return;

            this.isPeaceLover = value;
            OnPropertyChanged("IsPeaceLover", "IsEarthLover");
        }
    }

    public bool IsEarthLover
    {
        get { return !IsPeaceLover; }
        set { IsPeaceLover = !value; }
    }
    ...
}
If you have more options user can choose, you can extend the pattern like this:
public enum Priority
{
    Low,
    Middle,
    High,
}

class JobViewModel : ViewModelBase
{
    Priority priority = Priority.Low;

    public Priority Priority
    {
        get { return this.priority; }
        set
        {
            if (this.priority == value)
                return;

            this.priority = value;
            OnPropertyChanged("Priority"
                     , "IsLowPriority", "IsMiddlePriority", "IsHighPriority");
        }
    }

    public bool IsLowPriority
    {
        get { return Priority == Priority.Low; }
        set { Priority = value ? Priority.Low : Priority; }
    }

    public bool IsMiddlePriority
    {
        get { return Priority == Priority.Middle; }
        set { Priority = value ? Priority.Middle : Priority; }
    }

    public bool IsHighPriority
    {
        get { return Priority == Priority.High; }
        set { Priority = value ? Priority.High : Priority; }
    }
}
...
<GroupBox Header="Priority">
    <StackPanel>
        <RadioButton IsChecked="{Binding IsLowPriority}">Low</RadioButton>
        <RadioButton IsChecked="{Binding IsMiddlePriority}">Middle</RadioButton>
        <RadioButton IsChecked="{Binding IsHighPriority}">High</RadioButton>
    </StackPanel>
</GroupBox>
...


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