We have a small friends group using Telegram Messenger without any bot attached to group. When we send message in this group it is always seen (double checked) immediately even at midnight and hours that seems no one is online. Is there any security problem or any third party software attached to our group?
I think your group is in fact a so-called Super Group. Super Groups work a little differently than normal groups and the one-check/two-check system doesn't apply here. All messages get double checks instantly when received by the server. (That's a feature.)
The creator of your original group has clicked the 'upgrade to supergroup' button at some point, causing your group to upgrade. You cannot undo this upgrade, so if you want to go back to a normal group, you'll have to create a new one.
I'm guessing it has something to do with the original intend of Super Groups being aimed at large communities. These Super Groups were originally introduced for groups larger than 200 people, but since then the group and Super Group limits have been increased to 5000 members and upgrading to Super Groups is unlocked for Groups of any size.
Here is the original release blog post: https://telegram.org/blog/supergroups