Let's see what's causing the problem! Are you using a downloaded messaging application? Do you experience the issue with all recipients?
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I had the same problem with two phones - never fixed it with the first phone because it didn't bother me but with the new phone I had to have group messaging. I knew it was not my phone since the problem existed on my other older phone (same number, hence the problem was with Verizon and the line). SKIP customer support and the Verizon Store where you bought the phone.
Go to your HOME PHONE or another line - call 1-800-922-0204 ask for tech support (3), not customer service. Then when the person answers ASK if they are tech support (probably not) - ASK to be directed to TECH support.
Then have TECH SUPPORT "reset the messaging in their end switch" - you will need to power your mobile off. Power back on and you can group message - worked immediately. Good luck.